Min-max theorem - Wikipedia

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data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Variational characterization of eigenvalues of compact Hermitian operators on Hilbert spaces</div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Not to be confused with <a href="/wiki/Minimax_theorem" title="Minimax theorem">Minimax theorem</a>.</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">"Variational theorem" redirects here. Not to be confused with <a href="/wiki/Variational_principle" title="Variational principle">variational principle</a>.</div> <style data-mw-deduplicate="TemplateStyles:r1251242444">.mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px solid #36c;background-color:#fbfbfb;box-sizing:border-box}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+style+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+link+.ambox{margin-top:-1px}html body.mediawiki .mw-parser-output .ambox.mbox-small-left{margin:4px 1em 4px 0;overflow:hidden;width:238px;border-collapse:collapse;font-size:88%;line-height:1.25em}.mw-parser-output .ambox-speedy{border-left:10px solid #b32424;background-color:#fee7e6}.mw-parser-output .ambox-delete{border-left:10px solid #b32424}.mw-parser-output .ambox-content{border-left:10px solid #f28500}.mw-parser-output .ambox-style{border-left:10px solid #fc3}.mw-parser-output .ambox-move{border-left:10px solid #9932cc}.mw-parser-output .ambox-protection{border-left:10px solid #a2a9b1}.mw-parser-output .ambox .mbox-text{border:none;padding:0.25em 0.5em;width:100%}.mw-parser-output .ambox .mbox-image{border:none;padding:2px 0 2px 0.5em;text-align:center}.mw-parser-output .ambox .mbox-imageright{border:none;padding:2px 0.5em 2px 0;text-align:center}.mw-parser-output .ambox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .ambox .mbox-image-div{width:52px}@media(min-width:720px){.mw-parser-output .ambox{margin:0 10%}}@media print{body.ns-0 .mw-parser-output .ambox{display:none!important}}</style><table class="box-More_citations_needed plainlinks metadata ambox ambox-content ambox-Refimprove" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><a href="/wiki/File:Question_book-new.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/50px-Question_book-new.svg.png" decoding="async" width="50" height="39" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/75px-Question_book-new.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/100px-Question_book-new.svg.png 2x" data-file-width="512" data-file-height="399" /></a></span></div></td><td class="mbox-text"><div class="mbox-text-span">This article <b>needs additional citations for <a href="/wiki/Wikipedia:Verifiability" title="Wikipedia:Verifiability">verification</a></b>.<span class="hide-when-compact"> Please help <a href="/wiki/Special:EditPage/Min-max_theorem" title="Special:EditPage/Min-max theorem">improve this article</a> by <a href="/wiki/Help:Referencing_for_beginners" title="Help:Referencing for beginners">adding citations to reliable sources</a>. Unsourced material may be challenged and removed.<br /><small><span class="plainlinks"><i>Find sources:</i> <a rel="nofollow" class="external text" href="https://www.google.com/search?as_eq=wikipedia&q=%22Min-max+theorem%22">"Min-max theorem"</a> – <a rel="nofollow" class="external text" href="https://www.google.com/search?tbm=nws&q=%22Min-max+theorem%22+-wikipedia&tbs=ar:1">news</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.google.com/search?&q=%22Min-max+theorem%22&tbs=bkt:s&tbm=bks">newspapers</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.google.com/search?tbs=bks:1&q=%22Min-max+theorem%22+-wikipedia">books</a> <b>·</b> <a rel="nofollow" class="external text" href="https://scholar.google.com/scholar?q=%22Min-max+theorem%22">scholar</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.jstor.org/action/doBasicSearch?Query=%22Min-max+theorem%22&acc=on&wc=on">JSTOR</a></span></small></span> <span class="date-container"><i>(<span class="date">November 2011</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <p>In <a href="/wiki/Linear_algebra" title="Linear algebra">linear algebra</a> and <a href="/wiki/Functional_analysis" title="Functional analysis">functional analysis</a>, the <b>min-max theorem</b>, or <b>variational theorem</b>, or <b>Courant–Fischer–Weyl min-max principle</b>, is a result that gives a variational characterization of <a href="/wiki/Eigenvalues_and_eigenvectors" title="Eigenvalues and eigenvectors">eigenvalues</a> of <a href="/wiki/Compact_operator_on_Hilbert_space" title="Compact operator on Hilbert space">compact</a> Hermitian operators on <a href="/wiki/Hilbert_spaces" class="mw-redirect" title="Hilbert spaces">Hilbert spaces</a>. It can be viewed as the starting point of many results of similar nature. </p><p>This article first discusses the finite-dimensional case and its applications before considering compact operators on infinite-dimensional Hilbert spaces. We will see that for compact operators, the proof of the main theorem uses essentially the same idea from the finite-dimensional argument. </p><p>In the case that the operator is non-Hermitian, the theorem provides an equivalent characterization of the associated <a href="/wiki/Singular_values" class="mw-redirect" title="Singular values">singular values</a>. The min-max theorem can be extended to <a href="/wiki/Self-adjoint_operator" title="Self-adjoint operator">self-adjoint operators</a> that are bounded below. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Matrices">Matrices</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Min-max_theorem&action=edit&section=1" title="Edit section: Matrices"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Let <span class="texhtml mvar" style="font-style:italic;">A</span> be a <span class="texhtml"><i>n</i> × <i>n</i></span> <a href="/wiki/Hermitian_matrix" title="Hermitian matrix">Hermitian matrix</a>. As with many other variational results on eigenvalues, one considers the <a href="/wiki/Rayleigh_quotient" title="Rayleigh quotient">Rayleigh–Ritz quotient</a> <span class="texhtml"><i>R<sub>A</sub></i> : <b>C</b><sup><i>n</i></sup> \ {0} → <b>R</b></span> defined by </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{A}(x)={\frac {(Ax,x)}{(x,x)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>A</mi> <mi>x</mi> <mo>,</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{A}(x)={\frac {(Ax,x)}{(x,x)}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc03b154fba42316af2eef7a166cff58510ceee6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:17.548ex; height:6.509ex;" alt="{\displaystyle R_{A}(x)={\frac {(Ax,x)}{(x,x)}}}"></span></dd></dl> <p>where <span class="texhtml">(⋅, ⋅)</span> denotes the <a href="/wiki/Dot_product" title="Dot product">Euclidean inner product</a> on <span class="texhtml"><b>C</b><sup><i>n</i></sup></span>. Clearly, the Rayleigh quotient of an eigenvector is its associated eigenvalue. Equivalently, the Rayleigh–Ritz quotient can be replaced by </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)=(Ax,x),\;\|x\|=1.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mi>x</mi> <mo>,</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mspace width="thickmathspace" /> <mo fence="false" stretchy="false">‖</mo> <mi>x</mi> <mo fence="false" stretchy="false">‖</mo> <mo>=</mo> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)=(Ax,x),\;\|x\|=1.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b318917f2442042a02bd036f352b8a2a5392e76" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.003ex; height:2.843ex;" alt="{\displaystyle f(x)=(Ax,x),\;\|x\|=1.}"></span></dd></dl> <p>For Hermitian matrices <i>A</i>, the range of the continuous function <i>R<sub>A</sub></i>(<i>x</i>), or <i>f</i>(<i>x</i>), is a compact interval [<i>a</i>, <i>b</i>] of the real line. The maximum <i>b</i> and the minimum <i>a</i> are the largest and smallest eigenvalue of <i>A</i>, respectively. The min-max theorem is a refinement of this fact. </p> <div class="mw-heading mw-heading3"><h3 id="Min-max_theorem">Min-max theorem</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Min-max_theorem&action=edit&section=2" title="Edit section: Min-max theorem"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a118c6ad00742b3f5dccd2f0e74b5e369df6fd31" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\textstyle A}"></span> be Hermitian on an inner product space <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d67b50b7ba03a56fea637093cf80e12807852d19" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\textstyle V}"></span> with dimension <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc6e1f880981346a604257ebcacdef24c0aca2d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\textstyle n}"></span>, with spectrum ordered in descending order <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \lambda _{1}\geq ...\geq \lambda _{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>≥</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>≥</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \lambda _{1}\geq ...\geq \lambda _{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b6ad4253c70444a9335118d3c4fed137922365f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.992ex; height:2.509ex;" alt="{\textstyle \lambda _{1}\geq ...\geq \lambda _{n}}"></span>. </p><p>Let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle v_{1},...,v_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle v_{1},...,v_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d1cc5ed98dbf25d60b025abbd4d5a232edcd502" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.698ex; height:2.009ex;" alt="{\textstyle v_{1},...,v_{n}}"></span> be the corresponding unit-length orthogonal eigenvectors. </p><p>Reverse the spectrum ordering, so that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \xi _{1}=\lambda _{n},...,\xi _{n}=\lambda _{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <msub> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \xi _{1}=\lambda _{n},...,\xi _{n}=\lambda _{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/efe2f55bf67894684a83c077619105f6a4a102d5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:20.659ex; height:2.509ex;" alt="{\textstyle \xi _{1}=\lambda _{n},...,\xi _{n}=\lambda _{1}}"></span>. </p> <style data-mw-deduplicate="TemplateStyles:r1110004140">.mw-parser-output .math_theorem{margin:1em 2em;padding:0.5em 1em 0.4em;border:1px solid #aaa;overflow:hidden}@media(max-width:500px){.mw-parser-output .math_theorem{margin:1em 0em;padding:0.5em 0.5em 0.4em}}</style><div class="math_theorem" style=""> <p><strong class="theorem-name">(Poincaré’s inequality)</strong><span class="theoreme-tiret"> — </span>Let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/913ace920108f7552777e36ac0b7ee3f5093a088" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\textstyle M}"></span> be a subspace of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d67b50b7ba03a56fea637093cf80e12807852d19" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\textstyle V}"></span> with dimension <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d5595fc0c47452f8fc2aa6e29c3611f047714b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\textstyle k}"></span>, then there exists unit vectors <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle x,y\in M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>∈</mo> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle x,y\in M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/570e6a5fa71bfbec225c39be95f95a34ac84ada7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.802ex; height:2.509ex;" alt="{\textstyle x,y\in M}"></span>, such that </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \langle x,Ax\rangle \leq \lambda _{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>x</mi> <mo>,</mo> <mi>A</mi> <mi>x</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>≤</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \langle x,Ax\rangle \leq \lambda _{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c52bf00e26001e47bd7575486d5476fbba844491" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.788ex; height:2.843ex;" alt="{\textstyle \langle x,Ax\rangle \leq \lambda _{k}}"></span>, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \langle y,Ay\rangle \geq \xi _{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>y</mi> <mo>,</mo> <mi>A</mi> <mi>y</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>≥</mo> <msub> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \langle y,Ay\rangle \geq \xi _{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e409a6ba7465218cf7453ae89da4be44b5723543" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.103ex; height:2.843ex;" alt="{\textstyle \langle y,Ay\rangle \geq \xi _{k}}"></span>. </p> </div> <style data-mw-deduplicate="TemplateStyles:r1174254338">.mw-parser-output .math_proof{border:thin solid #aaa;margin:1em 2em;padding:0.5em 1em 0.4em}@media(max-width:500px){.mw-parser-output .math_proof{margin:1em 0;padding:0.5em 0.5em 0.4em}}</style><div class="math_proof" style=""><strong>Proof</strong> <p>Part 2 is a corollary, using <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle -A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo>−</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle -A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f49ab44fb42f2fe76da25505e9b1188a57654501" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:3.551ex; height:2.343ex;" alt="{\textstyle -A}"></span>. </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/913ace920108f7552777e36ac0b7ee3f5093a088" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\textstyle M}"></span> is a <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d5595fc0c47452f8fc2aa6e29c3611f047714b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\textstyle k}"></span> dimensional subspace, so if we pick any list of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle n-k+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>n</mi> <mo>−</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle n-k+1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/79aa5c1103760c13f146b1d67726ccbdb402d20c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.449ex; height:2.343ex;" alt="{\textstyle n-k+1}"></span> vectors, their span <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle N:=span(v_{k},...v_{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>N</mi> <mo>:=</mo> <mi>s</mi> <mi>p</mi> <mi>a</mi> <mi>n</mi> <mo stretchy="false">(</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle N:=span(v_{k},...v_{n})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b1ae45962499329a10158ca14c9ca1cc2b46067e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.201ex; height:2.843ex;" alt="{\textstyle N:=span(v_{k},...v_{n})}"></span> must intersect <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/913ace920108f7552777e36ac0b7ee3f5093a088" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\textstyle M}"></span> on at least a single line. </p><p>Take unit <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle x\in M\cap N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>x</mi> <mo>∈</mo> <mi>M</mi> <mo>∩</mo> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle x\in M\cap N}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/71d1aa220f751f5e052996b55bfa5ed665079dc4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.259ex; height:2.176ex;" alt="{\textstyle x\in M\cap N}"></span>. That’s what we need. </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle x=\sum _{i=k}^{n}a_{i}v_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mi>k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle x=\sum _{i=k}^{n}a_{i}v_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b1b28194bdbc9f3692b0330386834528e34f01c7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.161ex; height:3.176ex;" alt="{\textstyle x=\sum _{i=k}^{n}a_{i}v_{i}}"></span>, since <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle x\in N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>x</mi> <mo>∈</mo> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle x\in N}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db883c7aefff0bbc4ace7a0c4d29eedffc2bea50" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.234ex; height:2.176ex;" alt="{\textstyle x\in N}"></span>.</dd></dl> <dl><dd>Since <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \sum _{i=k}^{n}|a_{i}|^{2}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mi>k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \sum _{i=k}^{n}|a_{i}|^{2}=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7221f3f77d3c24e1663f0dbd5cd57a6e3cec0d08" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.414ex; height:3.509ex;" alt="{\textstyle \sum _{i=k}^{n}|a_{i}|^{2}=1}"></span>, we find <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \langle x,Ax\rangle =\sum _{i=k}^{n}|a_{i}|^{2}\lambda _{i}\leq \lambda _{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>x</mi> <mo>,</mo> <mi>A</mi> <mi>x</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mi>k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>≤</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \langle x,Ax\rangle =\sum _{i=k}^{n}|a_{i}|^{2}\lambda _{i}\leq \lambda _{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3a3663e91ff6e1be4e1f490b896de257f3103c55" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:28.195ex; height:3.509ex;" alt="{\textstyle \langle x,Ax\rangle =\sum _{i=k}^{n}|a_{i}|^{2}\lambda _{i}\leq \lambda _{k}}"></span>.</dd></dl> </div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1110004140"><div class="math_theorem" style=""> <p><strong class="theorem-name">min-max theorem</strong><span class="theoreme-tiret"> — </span><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\lambda _{k}&=\max _{\begin{array}{c}{\mathcal {M}}\subset V\\\operatorname {dim} ({\mathcal {M}})=k\end{array}}\min _{\begin{array}{c}x\in {\mathcal {M}}\\\|x\|=1\end{array}}\langle x,Ax\rangle \\&=\min _{\begin{array}{c}{\mathcal {M}}\subset V\\\operatorname {dim} ({\mathcal {M}})=n-k+1\end{array}}\max _{\begin{array}{c}x\in {\mathcal {M}}\\\|x\|=1\end{array}}\langle x,Ax\rangle {\text{. }}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">max</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">M</mi> </mrow> </mrow> <mo>⊂</mo> <mi>V</mi> </mtd> </mtr> <mtr> <mtd> <mi>dim</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">M</mi> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mi>k</mi> </mtd> </mtr> </mtable> </mrow> </munder> <munder> <mo movablelimits="true" form="prefix">min</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>x</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">M</mi> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mo fence="false" stretchy="false">‖</mo> <mi>x</mi> <mo fence="false" stretchy="false">‖</mo> <mo>=</mo> <mn>1</mn> </mtd> </mtr> </mtable> </mrow> </munder> <mo fence="false" stretchy="false">⟨</mo> <mi>x</mi> <mo>,</mo> <mi>A</mi> <mi>x</mi> <mo fence="false" stretchy="false">⟩</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">min</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">M</mi> </mrow> </mrow> <mo>⊂</mo> <mi>V</mi> </mtd> </mtr> <mtr> <mtd> <mi>dim</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">M</mi> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mi>n</mi> <mo>−</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mtd> </mtr> </mtable> </mrow> </munder> <munder> <mo movablelimits="true" form="prefix">max</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>x</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">M</mi> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mo fence="false" stretchy="false">‖</mo> <mi>x</mi> <mo fence="false" stretchy="false">‖</mo> <mo>=</mo> <mn>1</mn> </mtd> </mtr> </mtable> </mrow> </munder> <mo fence="false" stretchy="false">⟨</mo> <mi>x</mi> <mo>,</mo> <mi>A</mi> <mi>x</mi> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>. </mtext> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\lambda _{k}&=\max _{\begin{array}{c}{\mathcal {M}}\subset V\\\operatorname {dim} ({\mathcal {M}})=k\end{array}}\min _{\begin{array}{c}x\in {\mathcal {M}}\\\|x\|=1\end{array}}\langle x,Ax\rangle \\&=\min _{\begin{array}{c}{\mathcal {M}}\subset V\\\operatorname {dim} ({\mathcal {M}})=n-k+1\end{array}}\max _{\begin{array}{c}x\in {\mathcal {M}}\\\|x\|=1\end{array}}\langle x,Ax\rangle {\text{. }}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c68c952fcf12332d74e51d911bd44013bf9be02e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.838ex; width:33.835ex; height:14.843ex;" alt="{\displaystyle {\begin{aligned}\lambda _{k}&=\max _{\begin{array}{c}{\mathcal {M}}\subset V\\\operatorname {dim} ({\mathcal {M}})=k\end{array}}\min _{\begin{array}{c}x\in {\mathcal {M}}\\\|x\|=1\end{array}}\langle x,Ax\rangle \\&=\min _{\begin{array}{c}{\mathcal {M}}\subset V\\\operatorname {dim} ({\mathcal {M}})=n-k+1\end{array}}\max _{\begin{array}{c}x\in {\mathcal {M}}\\\|x\|=1\end{array}}\langle x,Ax\rangle {\text{. }}\end{aligned}}}"></span> </p> </div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1174254338"><div class="math_proof" style=""><strong>Proof</strong> <p>Part 2 is a corollary of part 1, by using <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle -A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo>−</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle -A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f49ab44fb42f2fe76da25505e9b1188a57654501" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:3.551ex; height:2.343ex;" alt="{\textstyle -A}"></span>. </p><p>By Poincare’s inequality, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \lambda _{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \lambda _{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6eb33199e6d4634c81ee0683035520ac2c5ed9ca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.444ex; height:2.509ex;" alt="{\textstyle \lambda _{k}}"></span> is an upper bound to the right side. </p><p>By setting <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\mathcal {M}}=span(v_{1},...v_{k})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">M</mi> </mrow> </mrow> <mo>=</mo> <mi>s</mi> <mi>p</mi> <mi>a</mi> <mi>n</mi> <mo stretchy="false">(</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\mathcal {M}}=span(v_{1},...v_{k})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f522feef04ca9e301d673ff9b5709541a67258ca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.117ex; height:2.843ex;" alt="{\textstyle {\mathcal {M}}=span(v_{1},...v_{k})}"></span>, the upper bound is achieved. </p> </div> <div class="mw-heading mw-heading3"><h3 id="Counterexample_in_the_non-Hermitian_case">Counterexample in the non-Hermitian case</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Min-max_theorem&action=edit&section=3" title="Edit section: Counterexample in the non-Hermitian case"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Let <i>N</i> be the nilpotent matrix </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}0&1\\0&0\end{bmatrix}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}0&1\\0&0\end{bmatrix}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c36baae7901efb976700cd8774e2665359d241ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:8.501ex; height:6.176ex;" alt="{\displaystyle {\begin{bmatrix}0&1\\0&0\end{bmatrix}}.}"></span></dd></dl> <p>Define the Rayleigh quotient <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{N}(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{N}(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/78a7c419196dc54bef6b34b348ace59c051a24ee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.594ex; height:2.843ex;" alt="{\displaystyle R_{N}(x)}"></span> exactly as above in the Hermitian case. Then it is easy to see that the only eigenvalue of <i>N</i> is zero, while the maximum value of the Rayleigh quotient is <span class="texhtml"><style data-mw-deduplicate="TemplateStyles:r1214402035">.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}</style><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span></span>. That is, the maximum value of the Rayleigh quotient is larger than the maximum eigenvalue. </p> <div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Min-max_theorem&action=edit&section=4" title="Edit section: Applications"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Min-max_principle_for_singular_values">Min-max principle for singular values</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Min-max_theorem&action=edit&section=5" title="Edit section: Min-max principle for singular values"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Singular_value" title="Singular value">singular values</a> {<i>σ<sub>k</sub></i>} of a square matrix <i>M</i> are the square roots of the eigenvalues of <i>M</i>*<i>M</i> (equivalently <i>MM*</i>). An immediate consequence<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="claim is unreferenced and maybe suspicious (April 2014)">citation needed</span></a></i>]</sup> of the first equality in the min-max theorem is: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{k}^{\downarrow }=\max _{S:\dim(S)=k}\min _{x\in S,\|x\|=1}(M^{*}Mx,x)^{\frac {1}{2}}=\max _{S:\dim(S)=k}\min _{x\in S,\|x\|=1}\|Mx\|.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">↓</mo> </mrow> </msubsup> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">max</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> <mo>:</mo> <mi>dim</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>S</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>k</mi> </mrow> </munder> <munder> <mo movablelimits="true" form="prefix">min</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>∈</mo> <mi>S</mi> <mo>,</mo> <mo fence="false" stretchy="false">‖</mo> <mi>x</mi> <mo fence="false" stretchy="false">‖</mo> <mo>=</mo> <mn>1</mn> </mrow> </munder> <mo stretchy="false">(</mo> <msup> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mi>M</mi> <mi>x</mi> <mo>,</mo> <mi>x</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </msup> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">max</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> <mo>:</mo> <mi>dim</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>S</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>k</mi> </mrow> </munder> <munder> <mo movablelimits="true" form="prefix">min</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>∈</mo> <mi>S</mi> <mo>,</mo> <mo fence="false" stretchy="false">‖</mo> <mi>x</mi> <mo fence="false" stretchy="false">‖</mo> <mo>=</mo> <mn>1</mn> </mrow> </munder> <mo fence="false" stretchy="false">‖</mo> <mi>M</mi> <mi>x</mi> <mo fence="false" stretchy="false">‖</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{k}^{\downarrow }=\max _{S:\dim(S)=k}\min _{x\in S,\|x\|=1}(M^{*}Mx,x)^{\frac {1}{2}}=\max _{S:\dim(S)=k}\min _{x\in S,\|x\|=1}\|Mx\|.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d42895c98a7513fc5091061debfade994e2591d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:63.693ex; height:5.676ex;" alt="{\displaystyle \sigma _{k}^{\downarrow }=\max _{S:\dim(S)=k}\min _{x\in S,\|x\|=1}(M^{*}Mx,x)^{\frac {1}{2}}=\max _{S:\dim(S)=k}\min _{x\in S,\|x\|=1}\|Mx\|.}"></span></dd></dl> <p>Similarly, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{k}^{\downarrow }=\min _{S:\dim(S)=n-k+1}\max _{x\in S,\|x\|=1}\|Mx\|.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">↓</mo> </mrow> </msubsup> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">min</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> <mo>:</mo> <mi>dim</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>S</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>n</mi> <mo>−</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </munder> <munder> <mo movablelimits="true" form="prefix">max</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>∈</mo> <mi>S</mi> <mo>,</mo> <mo fence="false" stretchy="false">‖</mo> <mi>x</mi> <mo fence="false" stretchy="false">‖</mo> <mo>=</mo> <mn>1</mn> </mrow> </munder> <mo fence="false" stretchy="false">‖</mo> <mi>M</mi> <mi>x</mi> <mo fence="false" stretchy="false">‖</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{k}^{\downarrow }=\min _{S:\dim(S)=n-k+1}\max _{x\in S,\|x\|=1}\|Mx\|.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6bef0c507f13beccbf3a0d8dc0f7860a145f66f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:34.369ex; height:5.009ex;" alt="{\displaystyle \sigma _{k}^{\downarrow }=\min _{S:\dim(S)=n-k+1}\max _{x\in S,\|x\|=1}\|Mx\|.}"></span></dd></dl> <p>Here <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{k}^{\downarrow }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">↓</mo> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{k}^{\downarrow }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a09c2a2ee7f6ce9e88ff7be991b6dc1f32c725c0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.416ex; height:3.509ex;" alt="{\displaystyle \sigma _{k}^{\downarrow }}"></span> denotes the <i>k</i><sup>th</sup> entry in the decreasing sequence of the singular values, so that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{1}^{\downarrow }\geq \sigma _{2}^{\downarrow }\geq \cdots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">↓</mo> </mrow> </msubsup> <mo>≥</mo> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">↓</mo> </mrow> </msubsup> <mo>≥</mo> <mo>⋯</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{1}^{\downarrow }\geq \sigma _{2}^{\downarrow }\geq \cdots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e773352c547e7b24d7d3e0e94650a679abe2afed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.69ex; height:3.509ex;" alt="{\displaystyle \sigma _{1}^{\downarrow }\geq \sigma _{2}^{\downarrow }\geq \cdots }"></span>. </p> <div class="mw-heading mw-heading3"><h3 id="Cauchy_interlacing_theorem">Cauchy interlacing theorem</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Min-max_theorem&action=edit&section=6" title="Edit section: Cauchy interlacing theorem"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Poincar%C3%A9_separation_theorem" title="Poincaré separation theorem">Poincaré separation theorem</a></div> <p>Let <span class="texhtml mvar" style="font-style:italic;">A</span> be a symmetric <i>n</i> × <i>n</i> matrix. The <i>m</i> × <i>m</i> matrix <i>B</i>, where <i>m</i> ≤ <i>n</i>, is called a <b><a href="/wiki/Compression_(functional_analysis)" title="Compression (functional analysis)">compression</a></b> of <span class="texhtml mvar" style="font-style:italic;">A</span> if there exists an <a href="/wiki/Projection_(linear_algebra)#Orthogonal_projections" title="Projection (linear algebra)">orthogonal projection</a> <i>P</i> onto a subspace of dimension <i>m</i> such that <i>PAP*</i> = <i>B</i>. The Cauchy interlacing theorem states: </p> <dl><dd><b>Theorem.</b> If the eigenvalues of <span class="texhtml mvar" style="font-style:italic;">A</span> are <span class="texhtml"><i>α</i><sub>1</sub> ≤ ... ≤ <i>α<sub>n</sub></i></span>, and those of <i>B</i> are <span class="texhtml"><i>β</i><sub>1</sub> ≤ ... ≤ <i>β<sub>j</sub></i> ≤ ... ≤ <i>β<sub>m</sub></i></span>, then for all <span class="texhtml"><i>j</i> ≤ <i>m</i></span>, <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha _{j}\leq \beta _{j}\leq \alpha _{n-m+j}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>≤</mo> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>≤</mo> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mi>m</mi> <mo>+</mo> <mi>j</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha _{j}\leq \beta _{j}\leq \alpha _{n-m+j}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/78356b0fdd006c0b809ee94d2a84092988cef4f4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:18.85ex; height:2.843ex;" alt="{\displaystyle \alpha _{j}\leq \beta _{j}\leq \alpha _{n-m+j}.}"></span></dd></dl></dd></dl> <p>This can be proven using the min-max principle. Let <i>β<sub>i</sub></i> have corresponding eigenvector <i>b<sub>i</sub></i> and <i>S<sub>j</sub></i> be the <i>j</i> dimensional subspace <span class="texhtml"><i>S<sub>j</sub></i> = span{<i>b</i><sub>1</sub>, ..., <i>b<sub>j</sub></i>},</span> then </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta _{j}=\max _{x\in S_{j},\|x\|=1}(Bx,x)=\max _{x\in S_{j},\|x\|=1}(PAP^{*}x,x)\geq \min _{S_{j}}\max _{x\in S_{j},\|x\|=1}(A(P^{*}x),P^{*}x)=\alpha _{j}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">max</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>∈</mo> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>,</mo> <mo fence="false" stretchy="false">‖</mo> <mi>x</mi> <mo fence="false" stretchy="false">‖</mo> <mo>=</mo> <mn>1</mn> </mrow> </munder> <mo stretchy="false">(</mo> <mi>B</mi> <mi>x</mi> <mo>,</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">max</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>∈</mo> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>,</mo> <mo fence="false" stretchy="false">‖</mo> <mi>x</mi> <mo fence="false" stretchy="false">‖</mo> <mo>=</mo> <mn>1</mn> </mrow> </munder> <mo stretchy="false">(</mo> <mi>P</mi> <mi>A</mi> <msup> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mi>x</mi> <mo>,</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>≥</mo> <munder> <mo movablelimits="true" form="prefix">min</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> </munder> <munder> <mo movablelimits="true" form="prefix">max</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>∈</mo> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>,</mo> <mo fence="false" stretchy="false">‖</mo> <mi>x</mi> <mo fence="false" stretchy="false">‖</mo> <mo>=</mo> <mn>1</mn> </mrow> </munder> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">(</mo> <msup> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mi>x</mi> <mo stretchy="false">)</mo> <mo>,</mo> <msup> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta _{j}=\max _{x\in S_{j},\|x\|=1}(Bx,x)=\max _{x\in S_{j},\|x\|=1}(PAP^{*}x,x)\geq \min _{S_{j}}\max _{x\in S_{j},\|x\|=1}(A(P^{*}x),P^{*}x)=\alpha _{j}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a4e8008676e33225ce803ef4c6df7a77616a7f5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:82.567ex; height:4.676ex;" alt="{\displaystyle \beta _{j}=\max _{x\in S_{j},\|x\|=1}(Bx,x)=\max _{x\in S_{j},\|x\|=1}(PAP^{*}x,x)\geq \min _{S_{j}}\max _{x\in S_{j},\|x\|=1}(A(P^{*}x),P^{*}x)=\alpha _{j}.}"></span></dd></dl> <p>According to first part of min-max, <span class="texhtml"><i>α<sub>j</sub></i> ≤ <i>β<sub>j</sub></i>.</span> On the other hand, if we define <span class="texhtml"><i>S</i><sub><i>m</i>−<i>j</i>+1</sub> = span{<i>b<sub>j</sub></i>, ..., <i>b<sub>m</sub></i>},</span> then </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta _{j}=\min _{x\in S_{m-j+1},\|x\|=1}(Bx,x)=\min _{x\in S_{m-j+1},\|x\|=1}(PAP^{*}x,x)=\min _{x\in S_{m-j+1},\|x\|=1}(A(P^{*}x),P^{*}x)\leq \alpha _{n-m+j},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">min</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>∈</mo> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>−</mo> <mi>j</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo fence="false" stretchy="false">‖</mo> <mi>x</mi> <mo fence="false" stretchy="false">‖</mo> <mo>=</mo> <mn>1</mn> </mrow> </munder> <mo stretchy="false">(</mo> <mi>B</mi> <mi>x</mi> <mo>,</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">min</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>∈</mo> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>−</mo> <mi>j</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo fence="false" stretchy="false">‖</mo> <mi>x</mi> <mo fence="false" stretchy="false">‖</mo> <mo>=</mo> <mn>1</mn> </mrow> </munder> <mo stretchy="false">(</mo> <mi>P</mi> <mi>A</mi> <msup> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mi>x</mi> <mo>,</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">min</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>∈</mo> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>−</mo> <mi>j</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo fence="false" stretchy="false">‖</mo> <mi>x</mi> <mo fence="false" stretchy="false">‖</mo> <mo>=</mo> <mn>1</mn> </mrow> </munder> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">(</mo> <msup> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mi>x</mi> <mo stretchy="false">)</mo> <mo>,</mo> <msup> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mi>x</mi> <mo stretchy="false">)</mo> <mo>≤</mo> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mi>m</mi> <mo>+</mo> <mi>j</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta _{j}=\min _{x\in S_{m-j+1},\|x\|=1}(Bx,x)=\min _{x\in S_{m-j+1},\|x\|=1}(PAP^{*}x,x)=\min _{x\in S_{m-j+1},\|x\|=1}(A(P^{*}x),P^{*}x)\leq \alpha _{n-m+j},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/027ace0787270ae0cacb8bbedf95641b79266698" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:95.035ex; height:4.676ex;" alt="{\displaystyle \beta _{j}=\min _{x\in S_{m-j+1},\|x\|=1}(Bx,x)=\min _{x\in S_{m-j+1},\|x\|=1}(PAP^{*}x,x)=\min _{x\in S_{m-j+1},\|x\|=1}(A(P^{*}x),P^{*}x)\leq \alpha _{n-m+j},}"></span></dd></dl> <p>where the last inequality is given by the second part of min-max. </p><p>When <span class="texhtml"><i>n</i> − <i>m</i> = 1</span>, we have <span class="texhtml"><i>α<sub>j</sub></i> ≤ <i>β<sub>j</sub></i> ≤ <i>α</i><sub><i>j</i>+1</sub></span>, hence the name <i>interlacing</i> theorem. </p> <div class="mw-heading mw-heading2"><h2 id="Compact_operators">Compact operators</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Min-max_theorem&action=edit&section=7" title="Edit section: Compact operators"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Let <span class="texhtml mvar" style="font-style:italic;">A</span> be a <a href="/wiki/Compact_operator_on_Hilbert_space" title="Compact operator on Hilbert space">compact</a>, <a href="/wiki/Hermitian" class="mw-redirect" title="Hermitian">Hermitian</a> operator on a Hilbert space <i>H</i>. Recall that the <a href="/wiki/Spectrum_(functional_analysis)" title="Spectrum (functional analysis)">spectrum</a> of such an operator (the set of eigenvalues) is a set of real numbers whose only possible <a href="/wiki/Cluster_point" class="mw-redirect" title="Cluster point">cluster point</a> is zero. It is thus convenient to list the positive eigenvalues of <span class="texhtml mvar" style="font-style:italic;">A</span> as </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cdots \leq \lambda _{k}\leq \cdots \leq \lambda _{1},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⋯</mo> <mo>≤</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>≤</mo> <mo>⋯</mo> <mo>≤</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cdots \leq \lambda _{k}\leq \cdots \leq \lambda _{1},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/640d8f6357fa9ed73afe94fb51a33a31ca2b5c4f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:20.242ex; height:2.509ex;" alt="{\displaystyle \cdots \leq \lambda _{k}\leq \cdots \leq \lambda _{1},}"></span></dd></dl> <p>where entries are repeated with <a href="/wiki/Multiplicity_(mathematics)" title="Multiplicity (mathematics)">multiplicity</a>, as in the matrix case. (To emphasize that the sequence is decreasing, we may write <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda _{k}=\lambda _{k}^{\downarrow }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>=</mo> <msubsup> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">↓</mo> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda _{k}=\lambda _{k}^{\downarrow }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/446271bc68bd6604d48e39951c19bb70e9c47d46" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.986ex; height:3.509ex;" alt="{\displaystyle \lambda _{k}=\lambda _{k}^{\downarrow }}"></span>.) When <i>H</i> is infinite-dimensional, the above sequence of eigenvalues is necessarily infinite. We now apply the same reasoning as in the matrix case. Letting <i>S<sub>k</sub></i> ⊂ <i>H</i> be a <i>k</i> dimensional subspace, we can obtain the following theorem. </p> <dl><dd><b>Theorem (Min-Max).</b> Let <span class="texhtml mvar" style="font-style:italic;">A</span> be a compact, self-adjoint operator on a Hilbert space <span class="texhtml mvar" style="font-style:italic;">H</span>, whose positive eigenvalues are listed in decreasing order <span class="texhtml">... ≤ <i>λ<sub>k</sub></i> ≤ ... ≤ <i>λ</i><sub>1</sub></span>. Then: <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\max _{S_{k}}\min _{x\in S_{k},\|x\|=1}(Ax,x)&=\lambda _{k}^{\downarrow },\\\min _{S_{k-1}}\max _{x\in S_{k-1}^{\perp },\|x\|=1}(Ax,x)&=\lambda _{k}^{\downarrow }.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <munder> <mo movablelimits="true" form="prefix">max</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> </munder> <munder> <mo movablelimits="true" form="prefix">min</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>∈</mo> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>,</mo> <mo fence="false" stretchy="false">‖</mo> <mi>x</mi> <mo fence="false" stretchy="false">‖</mo> <mo>=</mo> <mn>1</mn> </mrow> </munder> <mo stretchy="false">(</mo> <mi>A</mi> <mi>x</mi> <mo>,</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <msubsup> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">↓</mo> </mrow> </msubsup> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <munder> <mo movablelimits="true" form="prefix">min</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mrow> </munder> <munder> <mo movablelimits="true" form="prefix">max</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>∈</mo> <msubsup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>⊥</mo> </mrow> </msubsup> <mo>,</mo> <mo fence="false" stretchy="false">‖</mo> <mi>x</mi> <mo fence="false" stretchy="false">‖</mo> <mo>=</mo> <mn>1</mn> </mrow> </munder> <mo stretchy="false">(</mo> <mi>A</mi> <mi>x</mi> <mo>,</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <msubsup> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">↓</mo> </mrow> </msubsup> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\max _{S_{k}}\min _{x\in S_{k},\|x\|=1}(Ax,x)&=\lambda _{k}^{\downarrow },\\\min _{S_{k-1}}\max _{x\in S_{k-1}^{\perp },\|x\|=1}(Ax,x)&=\lambda _{k}^{\downarrow }.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9984269e829e8bbb243ab2469e531979b655276" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.838ex; width:29.2ex; height:10.843ex;" alt="{\displaystyle {\begin{aligned}\max _{S_{k}}\min _{x\in S_{k},\|x\|=1}(Ax,x)&=\lambda _{k}^{\downarrow },\\\min _{S_{k-1}}\max _{x\in S_{k-1}^{\perp },\|x\|=1}(Ax,x)&=\lambda _{k}^{\downarrow }.\end{aligned}}}"></span></dd></dl></dd></dl> <p>A similar pair of equalities hold for negative eigenvalues. </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1174254338"><div class="math_proof" style=""><strong>Proof</strong> <p>Let <i>S' </i> be the closure of the linear span <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S'=\operatorname {span} \{u_{k},u_{k+1},\ldots \}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>S</mi> <mo>′</mo> </msup> <mo>=</mo> <mi>span</mi> <mo>⁡</mo> <mo fence="false" stretchy="false">{</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S'=\operatorname {span} \{u_{k},u_{k+1},\ldots \}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/844c11407db842ec6a507919101b6196ff6dd95a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.022ex; height:3.009ex;" alt="{\displaystyle S'=\operatorname {span} \{u_{k},u_{k+1},\ldots \}}"></span>. The subspace <i>S' </i> has codimension <i>k</i> − 1. By the same dimension count argument as in the matrix case, <i>S' </i> ∩ <i>S<sub>k</sub></i> has positive dimension. So there exists <i>x</i> ∈ <i>S' </i> ∩ <i>S<sub>k</sub></i> with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|x\|=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mi>x</mi> <mo fence="false" stretchy="false">‖</mo> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|x\|=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e3060a9705882ba484bebd7ece4b538bbee5bbe6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.916ex; height:2.843ex;" alt="{\displaystyle \|x\|=1}"></span>. Since it is an element of <i>S' </i>, such an <i>x</i> necessarily satisfy </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (Ax,x)\leq \lambda _{k}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>A</mi> <mi>x</mi> <mo>,</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>≤</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (Ax,x)\leq \lambda _{k}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9628b1a0ae315028a25b926f75811a54d8de600e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.435ex; height:2.843ex;" alt="{\displaystyle (Ax,x)\leq \lambda _{k}.}"></span></dd></dl> <p>Therefore, for all <i>S<sub>k</sub></i> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \inf _{x\in S_{k},\|x\|=1}(Ax,x)\leq \lambda _{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">inf</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>∈</mo> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>,</mo> <mo fence="false" stretchy="false">‖</mo> <mi>x</mi> <mo fence="false" stretchy="false">‖</mo> <mo>=</mo> <mn>1</mn> </mrow> </munder> <mo stretchy="false">(</mo> <mi>A</mi> <mi>x</mi> <mo>,</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>≤</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \inf _{x\in S_{k},\|x\|=1}(Ax,x)\leq \lambda _{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/367c482b0fe083457e0e53c5985dff090dd520f4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:21.834ex; height:4.509ex;" alt="{\displaystyle \inf _{x\in S_{k},\|x\|=1}(Ax,x)\leq \lambda _{k}}"></span></dd></dl> <p>But <span class="texhtml mvar" style="font-style:italic;">A</span> is compact, therefore the function <i>f</i>(<i>x</i>) = (<i>Ax</i>, <i>x</i>) is weakly continuous. Furthermore, any bounded set in <i>H</i> is weakly compact. This lets us replace the infimum by minimum: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \min _{x\in S_{k},\|x\|=1}(Ax,x)\leq \lambda _{k}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">min</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>∈</mo> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>,</mo> <mo fence="false" stretchy="false">‖</mo> <mi>x</mi> <mo fence="false" stretchy="false">‖</mo> <mo>=</mo> <mn>1</mn> </mrow> </munder> <mo stretchy="false">(</mo> <mi>A</mi> <mi>x</mi> <mo>,</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>≤</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \min _{x\in S_{k},\|x\|=1}(Ax,x)\leq \lambda _{k}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/886c2a905efd3971591f223acbe9e6dfd928fb37" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:22.481ex; height:4.509ex;" alt="{\displaystyle \min _{x\in S_{k},\|x\|=1}(Ax,x)\leq \lambda _{k}.}"></span></dd></dl> <p>So </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sup _{S_{k}}\min _{x\in S_{k},\|x\|=1}(Ax,x)\leq \lambda _{k}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">sup</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> </munder> <munder> <mo movablelimits="true" form="prefix">min</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>∈</mo> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>,</mo> <mo fence="false" stretchy="false">‖</mo> <mi>x</mi> <mo fence="false" stretchy="false">‖</mo> <mo>=</mo> <mn>1</mn> </mrow> </munder> <mo stretchy="false">(</mo> <mi>A</mi> <mi>x</mi> <mo>,</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>≤</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sup _{S_{k}}\min _{x\in S_{k},\|x\|=1}(Ax,x)\leq \lambda _{k}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f4b89d7a89656852c1adafab50e3443bb129f86f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:26.369ex; height:4.843ex;" alt="{\displaystyle \sup _{S_{k}}\min _{x\in S_{k},\|x\|=1}(Ax,x)\leq \lambda _{k}.}"></span></dd></dl> <p>Because equality is achieved when <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{k}=\operatorname {span} \{u_{1},\ldots ,u_{k}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>=</mo> <mi>span</mi> <mo>⁡</mo> <mo fence="false" stretchy="false">{</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{k}=\operatorname {span} \{u_{1},\ldots ,u_{k}\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/95ec4eaa25ae5892c4b29b577ddfff925fcb8648" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.581ex; height:2.843ex;" alt="{\displaystyle S_{k}=\operatorname {span} \{u_{1},\ldots ,u_{k}\}}"></span>, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \max _{S_{k}}\min _{x\in S_{k},\|x\|=1}(Ax,x)=\lambda _{k}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">max</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> </munder> <munder> <mo movablelimits="true" form="prefix">min</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>∈</mo> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>,</mo> <mo fence="false" stretchy="false">‖</mo> <mi>x</mi> <mo fence="false" stretchy="false">‖</mo> <mo>=</mo> <mn>1</mn> </mrow> </munder> <mo stretchy="false">(</mo> <mi>A</mi> <mi>x</mi> <mo>,</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \max _{S_{k}}\min _{x\in S_{k},\|x\|=1}(Ax,x)=\lambda _{k}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f7ff8bee95d3b0f192d352787eb725ad11deabd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:27.194ex; height:4.509ex;" alt="{\displaystyle \max _{S_{k}}\min _{x\in S_{k},\|x\|=1}(Ax,x)=\lambda _{k}.}"></span></dd></dl> <p>This is the first part of min-max theorem for compact self-adjoint operators. </p><p>Analogously, consider now a <span class="texhtml">(<i>k</i> − 1)</span>-dimensional subspace <i>S</i><sub><i>k</i>−1</sub>, whose the orthogonal complement is denoted by <i>S</i><sub><i>k</i>−1</sub><sup>⊥</sup>. If <i>S' </i> = span{<i>u</i><sub>1</sub>...<i>u<sub>k</sub></i>}, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S'\cap S_{k-1}^{\perp }\neq {0}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>S</mi> <mo>′</mo> </msup> <mo>∩</mo> <msubsup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>⊥</mo> </mrow> </msubsup> <mo>≠</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S'\cap S_{k-1}^{\perp }\neq {0}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d66579808d5b37960c0ec68abcc5e6b0908dda40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:14.311ex; height:3.343ex;" alt="{\displaystyle S'\cap S_{k-1}^{\perp }\neq {0}.}"></span></dd></dl> <p>So </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \exists x\in S_{k-1}^{\perp }\,\|x\|=1,(Ax,x)\geq \lambda _{k}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∃</mi> <mi>x</mi> <mo>∈</mo> <msubsup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>⊥</mo> </mrow> </msubsup> <mspace width="thinmathspace" /> <mo fence="false" stretchy="false">‖</mo> <mi>x</mi> <mo fence="false" stretchy="false">‖</mo> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mi>x</mi> <mo>,</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>≥</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \exists x\in S_{k-1}^{\perp }\,\|x\|=1,(Ax,x)\geq \lambda _{k}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c081176eb335c2b66eb8a0b44ee64491d47c2351" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:32.849ex; height:3.343ex;" alt="{\displaystyle \exists x\in S_{k-1}^{\perp }\,\|x\|=1,(Ax,x)\geq \lambda _{k}.}"></span></dd></dl> <p>This implies </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \max _{x\in S_{k-1}^{\perp },\|x\|=1}(Ax,x)\geq \lambda _{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">max</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>∈</mo> <msubsup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>⊥</mo> </mrow> </msubsup> <mo>,</mo> <mo fence="false" stretchy="false">‖</mo> <mi>x</mi> <mo fence="false" stretchy="false">‖</mo> <mo>=</mo> <mn>1</mn> </mrow> </munder> <mo stretchy="false">(</mo> <mi>A</mi> <mi>x</mi> <mo>,</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>≥</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \max _{x\in S_{k-1}^{\perp },\|x\|=1}(Ax,x)\geq \lambda _{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47c6463b9cb5ec458b08d2d14a30bc4c0972939d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:23.539ex; height:5.176ex;" alt="{\displaystyle \max _{x\in S_{k-1}^{\perp },\|x\|=1}(Ax,x)\geq \lambda _{k}}"></span></dd></dl> <p>where the compactness of <i>A</i> was applied. Index the above by the collection of <i>k-1</i>-dimensional subspaces gives </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \inf _{S_{k-1}}\max _{x\in S_{k-1}^{\perp },\|x\|=1}(Ax,x)\geq \lambda _{k}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">inf</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mrow> </munder> <munder> <mo movablelimits="true" form="prefix">max</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>∈</mo> <msubsup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>⊥</mo> </mrow> </msubsup> <mo>,</mo> <mo fence="false" stretchy="false">‖</mo> <mi>x</mi> <mo fence="false" stretchy="false">‖</mo> <mo>=</mo> <mn>1</mn> </mrow> </munder> <mo stretchy="false">(</mo> <mi>A</mi> <mi>x</mi> <mo>,</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>≥</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \inf _{S_{k-1}}\max _{x\in S_{k-1}^{\perp },\|x\|=1}(Ax,x)\geq \lambda _{k}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f6a1068bd3d7d3d960335f08b30aa275983800e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:28.146ex; height:5.176ex;" alt="{\displaystyle \inf _{S_{k-1}}\max _{x\in S_{k-1}^{\perp },\|x\|=1}(Ax,x)\geq \lambda _{k}.}"></span></dd></dl> <p>Pick <i>S</i><sub><i>k</i>−1</sub> = span{<i>u</i><sub>1</sub>, ..., <i>u</i><sub><i>k</i>−1</sub>} and we deduce </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \min _{S_{k-1}}\max _{x\in S_{k-1}^{\perp },\|x\|=1}(Ax,x)=\lambda _{k}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">min</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mrow> </munder> <munder> <mo movablelimits="true" form="prefix">max</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>∈</mo> <msubsup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>⊥</mo> </mrow> </msubsup> <mo>,</mo> <mo fence="false" stretchy="false">‖</mo> <mi>x</mi> <mo fence="false" stretchy="false">‖</mo> <mo>=</mo> <mn>1</mn> </mrow> </munder> <mo stretchy="false">(</mo> <mi>A</mi> <mi>x</mi> <mo>,</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \min _{S_{k-1}}\max _{x\in S_{k-1}^{\perp },\|x\|=1}(Ax,x)=\lambda _{k}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/79315b7388ada12790863ee735cd97d00d6dc05e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:28.448ex; height:5.176ex;" alt="{\displaystyle \min _{S_{k-1}}\max _{x\in S_{k-1}^{\perp },\|x\|=1}(Ax,x)=\lambda _{k}.}"></span></dd></dl> </div> <div class="mw-heading mw-heading2"><h2 id="Self-adjoint_operators">Self-adjoint operators</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Min-max_theorem&action=edit&section=8" title="Edit section: Self-adjoint operators"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The min-max theorem also applies to (possibly unbounded) self-adjoint operators.<sup id="cite_ref-teschl_1-0" class="reference"><a href="#cite_note-teschl-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-lieb-loss_2-0" class="reference"><a href="#cite_note-lieb-loss-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> Recall the <a href="/wiki/Essential_spectrum" title="Essential spectrum">essential spectrum</a> is the spectrum without isolated eigenvalues of finite multiplicity. Sometimes we have some eigenvalues below the essential spectrum, and we would like to approximate the eigenvalues and eigenfunctions. </p> <dl><dd><b>Theorem (Min-Max).</b> Let <i>A</i> be self-adjoint, and let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{1}\leq E_{2}\leq E_{3}\leq \cdots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>≤</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>≤</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>≤</mo> <mo>⋯</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{1}\leq E_{2}\leq E_{3}\leq \cdots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2864b61cefbde1634de8c6d7c4d11fbb8006eb57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:20.327ex; height:2.509ex;" alt="{\displaystyle E_{1}\leq E_{2}\leq E_{3}\leq \cdots }"></span> be the eigenvalues of <i>A</i> below the essential spectrum. Then</dd></dl> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{n}=\min _{\psi _{1},\ldots ,\psi _{n}}\max\{\langle \psi ,A\psi \rangle :\psi \in \operatorname {span} (\psi _{1},\ldots ,\psi _{n}),\,\|\psi \|=1\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">min</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </munder> <mo movablelimits="true" form="prefix">max</mo> <mo fence="false" stretchy="false">{</mo> <mo fence="false" stretchy="false">⟨</mo> <mi>ψ</mi> <mo>,</mo> <mi>A</mi> <mi>ψ</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>:</mo> <mi>ψ</mi> <mo>∈</mo> <mi>span</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>,</mo> <mspace width="thinmathspace" /> <mo fence="false" stretchy="false">‖</mo> <mi>ψ</mi> <mo fence="false" stretchy="false">‖</mo> <mo>=</mo> <mn>1</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{n}=\min _{\psi _{1},\ldots ,\psi _{n}}\max\{\langle \psi ,A\psi \rangle :\psi \in \operatorname {span} (\psi _{1},\ldots ,\psi _{n}),\,\|\psi \|=1\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5560be57176a3dba6f7cd29a5b48fbef6d706d85" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:60.221ex; height:4.509ex;" alt="{\displaystyle E_{n}=\min _{\psi _{1},\ldots ,\psi _{n}}\max\{\langle \psi ,A\psi \rangle :\psi \in \operatorname {span} (\psi _{1},\ldots ,\psi _{n}),\,\|\psi \|=1\}}"></span>. </p><p>If we only have <i>N</i> eigenvalues and hence run out of eigenvalues, then we let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{n}:=\inf \sigma _{ess}(A)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>:=</mo> <mo movablelimits="true" form="prefix">inf</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> <mi>s</mi> <mi>s</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{n}:=\inf \sigma _{ess}(A)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a64fbd25262dcc70fa473bf28dfebd9fb11b3971" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.138ex; height:2.843ex;" alt="{\displaystyle E_{n}:=\inf \sigma _{ess}(A)}"></span> (the bottom of the essential spectrum) for <i>n>N</i>, and the above statement holds after replacing min-max with inf-sup. </p> <dl><dd><b>Theorem (Max-Min).</b> Let <i>A</i> be self-adjoint, and let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{1}\leq E_{2}\leq E_{3}\leq \cdots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>≤</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>≤</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>≤</mo> <mo>⋯</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{1}\leq E_{2}\leq E_{3}\leq \cdots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2864b61cefbde1634de8c6d7c4d11fbb8006eb57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:20.327ex; height:2.509ex;" alt="{\displaystyle E_{1}\leq E_{2}\leq E_{3}\leq \cdots }"></span> be the eigenvalues of <i>A</i> below the essential spectrum. Then</dd></dl> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{n}=\max _{\psi _{1},\ldots ,\psi _{n-1}}\min\{\langle \psi ,A\psi \rangle :\psi \perp \psi _{1},\ldots ,\psi _{n-1},\,\|\psi \|=1\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">max</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mrow> </munder> <mo movablelimits="true" form="prefix">min</mo> <mo fence="false" stretchy="false">{</mo> <mo fence="false" stretchy="false">⟨</mo> <mi>ψ</mi> <mo>,</mo> <mi>A</mi> <mi>ψ</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>:</mo> <mi>ψ</mi> <mo>⊥</mo> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mspace width="thinmathspace" /> <mo fence="false" stretchy="false">‖</mo> <mi>ψ</mi> <mo fence="false" stretchy="false">‖</mo> <mo>=</mo> <mn>1</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{n}=\max _{\psi _{1},\ldots ,\psi _{n-1}}\min\{\langle \psi ,A\psi \rangle :\psi \perp \psi _{1},\ldots ,\psi _{n-1},\,\|\psi \|=1\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea963223fb913d0ccab636adc169b7dcb15d405a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:57.361ex; height:4.509ex;" alt="{\displaystyle E_{n}=\max _{\psi _{1},\ldots ,\psi _{n-1}}\min\{\langle \psi ,A\psi \rangle :\psi \perp \psi _{1},\ldots ,\psi _{n-1},\,\|\psi \|=1\}}"></span>. </p><p>If we only have <i>N</i> eigenvalues and hence run out of eigenvalues, then we let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{n}:=\inf \sigma _{ess}(A)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>:=</mo> <mo movablelimits="true" form="prefix">inf</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> <mi>s</mi> <mi>s</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{n}:=\inf \sigma _{ess}(A)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a64fbd25262dcc70fa473bf28dfebd9fb11b3971" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.138ex; height:2.843ex;" alt="{\displaystyle E_{n}:=\inf \sigma _{ess}(A)}"></span> (the bottom of the essential spectrum) for <i>n > N</i>, and the above statement holds after replacing max-min with sup-inf. </p><p>The proofs<sup id="cite_ref-teschl_1-1" class="reference"><a href="#cite_note-teschl-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-lieb-loss_2-1" class="reference"><a href="#cite_note-lieb-loss-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> use the following results about self-adjoint operators: </p> <dl><dd><b>Theorem.</b> Let <i>A</i> be self-adjoint. Then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (A-E)\geq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>A</mi> <mo>−</mo> <mi>E</mi> <mo stretchy="false">)</mo> <mo>≥</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (A-E)\geq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1904595179cf05029aa960dfca224614e59bb53a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.429ex; height:2.843ex;" alt="{\displaystyle (A-E)\geq 0}"></span> for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E\in \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E\in \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e204f8902c406c25b165a1db51773b7710ec38af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.294ex; height:2.176ex;" alt="{\displaystyle E\in \mathbb {R} }"></span> if and only if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma (A)\subseteq [E,\infty )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo>⊆</mo> <mo stretchy="false">[</mo> <mi>E</mi> <mo>,</mo> <mi mathvariant="normal">∞</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma (A)\subseteq [E,\infty )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9dae98326028146b5d2240ca75a4daa2b65d20d5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.665ex; height:2.843ex;" alt="{\displaystyle \sigma (A)\subseteq [E,\infty )}"></span>.<sup id="cite_ref-teschl_1-2" class="reference"><a href="#cite_note-teschl-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 77">: 77 </span></sup></dd></dl> <dl><dd><b>Theorem.</b> If <i>A</i> is self-adjoint, then</dd></dl> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \inf \sigma (A)=\inf _{\psi \in {\mathfrak {D}}(A),\|\psi \|=1}\langle \psi ,A\psi \rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">inf</mo> <mi>σ</mi> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">inf</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>ψ</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">D</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mo fence="false" stretchy="false">‖</mo> <mi>ψ</mi> <mo fence="false" stretchy="false">‖</mo> <mo>=</mo> <mn>1</mn> </mrow> </munder> <mo fence="false" stretchy="false">⟨</mo> <mi>ψ</mi> <mo>,</mo> <mi>A</mi> <mi>ψ</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \inf \sigma (A)=\inf _{\psi \in {\mathfrak {D}}(A),\|\psi \|=1}\langle \psi ,A\psi \rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e9e7a965d3341f41d93d91d08b589cfca3e45ece" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:29.949ex; height:4.509ex;" alt="{\displaystyle \inf \sigma (A)=\inf _{\psi \in {\mathfrak {D}}(A),\|\psi \|=1}\langle \psi ,A\psi \rangle }"></span> </p><p>and </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sup \sigma (A)=\sup _{\psi \in {\mathfrak {D}}(A),\|\psi \|=1}\langle \psi ,A\psi \rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">sup</mo> <mi>σ</mi> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">sup</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>ψ</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">D</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mo fence="false" stretchy="false">‖</mo> <mi>ψ</mi> <mo fence="false" stretchy="false">‖</mo> <mo>=</mo> <mn>1</mn> </mrow> </munder> <mo fence="false" stretchy="false">⟨</mo> <mi>ψ</mi> <mo>,</mo> <mi>A</mi> <mi>ψ</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sup \sigma (A)=\sup _{\psi \in {\mathfrak {D}}(A),\|\psi \|=1}\langle \psi ,A\psi \rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ed5124e87c47cc7cf76eb3741a40d8a039080d4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:30.799ex; height:5.009ex;" alt="{\displaystyle \sup \sigma (A)=\sup _{\psi \in {\mathfrak {D}}(A),\|\psi \|=1}\langle \psi ,A\psi \rangle }"></span>.<sup id="cite_ref-teschl_1-3" class="reference"><a href="#cite_note-teschl-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 77">: 77 </span></sup> </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Min-max_theorem&action=edit&section=9" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Courant_minimax_principle" title="Courant minimax principle">Courant minimax principle</a></li> <li><a href="/wiki/Max%E2%80%93min_inequality" title="Max–min inequality">Max–min inequality</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Min-max_theorem&action=edit&section=10" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-teschl-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-teschl_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-teschl_1-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-teschl_1-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-teschl_1-3"><sup><i><b>d</b></i></sup></a></span> <span class="reference-text">G. Teschl, Mathematical Methods in Quantum Mechanics (GSM 99) <a rel="nofollow" class="external free" href="https://www.mat.univie.ac.at/~gerald/ftp/book-schroe/schroe.pdf">https://www.mat.univie.ac.at/~gerald/ftp/book-schroe/schroe.pdf</a></span> </li> <li id="cite_note-lieb-loss-2"><span class="mw-cite-backlink">^ <a href="#cite_ref-lieb-loss_2-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-lieb-loss_2-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFLiebLoss2001" class="citation book cs1">Lieb; Loss (2001). <i>Analysis</i>. GSM. Vol. 14 (2nd ed.). Providence: American Mathematical Society. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-8218-2783-9" title="Special:BookSources/0-8218-2783-9"><bdi>0-8218-2783-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Analysis&rft.place=Providence&rft.series=GSM&rft.edition=2nd&rft.pub=American+Mathematical+Society&rft.date=2001&rft.isbn=0-8218-2783-9&rft.au=Lieb&rft.au=Loss&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMin-max+theorem" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="External_links_and_citations_to_related_work">External links and citations to related work</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Min-max_theorem&action=edit&section=11" title="Edit section: External links and citations to related work"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFisk2005" class="citation arxiv cs1">Fisk, Steve (2005). "A very short proof of Cauchy's interlace theorem for eigenvalues of Hermitian matrices". <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/math/0502408">math/0502408</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=preprint&rft.jtitle=arXiv&rft.atitle=A+very+short+proof+of+Cauchy%27s+interlace+theorem+for+eigenvalues+of+Hermitian+matrices&rft.date=2005&rft_id=info%3Aarxiv%2Fmath%2F0502408&rft.aulast=Fisk&rft.aufirst=Steve&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMin-max+theorem" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHwang2004" class="citation journal cs1">Hwang, Suk-Geun (2004). <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/4145217">"Cauchy's Interlace Theorem for Eigenvalues of Hermitian Matrices"</a>. <i>The American Mathematical Monthly</i>. <b>111</b> (2): 157–159. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F4145217">10.2307/4145217</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/4145217">4145217</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+American+Mathematical+Monthly&rft.atitle=Cauchy%27s+Interlace+Theorem+for+Eigenvalues+of+Hermitian+Matrices&rft.volume=111&rft.issue=2&rft.pages=157-159&rft.date=2004&rft_id=info%3Adoi%2F10.2307%2F4145217&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F4145217%23id-name%3DJSTOR&rft.aulast=Hwang&rft.aufirst=Suk-Geun&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F4145217&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMin-max+theorem" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKline2020" class="citation journal cs1">Kline, Jeffery (2020). <a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.laa.2019.12.004">"Bordered Hermitian matrices and sums of the Möbius function"</a>. <i>Linear Algebra and Its Applications</i>. <b>588</b>: 224–237. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.laa.2019.12.004">10.1016/j.laa.2019.12.004</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Linear+Algebra+and+Its+Applications&rft.atitle=Bordered+Hermitian+matrices+and+sums+of+the+M%C3%B6bius+function&rft.volume=588&rft.pages=224-237&rft.date=2020&rft_id=info%3Adoi%2F10.1016%2Fj.laa.2019.12.004&rft.aulast=Kline&rft.aufirst=Jeffery&rft_id=https%3A%2F%2Fdoi.org%2F10.1016%252Fj.laa.2019.12.004&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMin-max+theorem" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFReedSimon1978" class="citation book cs1">Reed, Michael; Simon, Barry (1978). <a rel="nofollow" class="external text" href="https://www.elsevier.com/books/iv-analysis-of-operators/reed/978-0-08-057045-7"><i>Methods of Modern Mathematical Physics IV: Analysis of Operators</i></a>. Academic Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-08-057045-7" title="Special:BookSources/978-0-08-057045-7"><bdi>978-0-08-057045-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Methods+of+Modern+Mathematical+Physics+IV%3A+Analysis+of+Operators&rft.pub=Academic+Press&rft.date=1978&rft.isbn=978-0-08-057045-7&rft.aulast=Reed&rft.aufirst=Michael&rft.au=Simon%2C+Barry&rft_id=https%3A%2F%2Fwww.elsevier.com%2Fbooks%2Fiv-analysis-of-operators%2Freed%2F978-0-08-057045-7&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMin-max+theorem" class="Z3988"></span></li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist 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style="width:1%">Spaces</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Banach_space" title="Banach space">Banach</a></li> <li><a href="/wiki/Besov_space" title="Besov space">Besov</a></li> <li><a href="/wiki/Fr%C3%A9chet_space" title="Fréchet space">Fréchet</a></li> <li><a href="/wiki/Hilbert_space" title="Hilbert space">Hilbert</a></li> <li><a href="/wiki/H%C3%B6lder_space" class="mw-redirect" title="Hölder space">Hölder</a></li> <li><a href="/wiki/Nuclear_space" title="Nuclear space">Nuclear</a></li> <li><a href="/wiki/Orlicz_space" title="Orlicz space">Orlicz</a></li> <li><a href="/wiki/Schwartz_space" title="Schwartz space">Schwartz</a></li> <li><a href="/wiki/Sobolev_space" title="Sobolev space">Sobolev</a></li> <li><a href="/wiki/Topological_vector_space" title="Topological vector space">Topological vector</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Properties</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Barrelled_space" title="Barrelled space">Barrelled</a></li> <li><a href="/wiki/Complete_topological_vector_space" title="Complete topological vector space">Complete</a></li> <li><a href="/wiki/Dual_space" title="Dual space">Dual</a> (<a href="/wiki/Dual_space#Algebraic_dual_space" title="Dual space">Algebraic</a>/<a href="/wiki/Dual_space#Continuous_dual_space" title="Dual space">Topological</a>)</li> <li><a href="/wiki/Locally_convex_topological_vector_space" title="Locally convex topological vector space">Locally convex</a></li> <li><a href="/wiki/Reflexive_space" title="Reflexive space">Reflexive</a></li> <li><a href="/wiki/Separable_space" title="Separable space">Separable</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Category:Theorems_in_functional_analysis" title="Category:Theorems in functional analysis">Theorems</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Hahn%E2%80%93Banach_theorem" title="Hahn–Banach theorem">Hahn–Banach</a></li> <li><a href="/wiki/Riesz_representation_theorem" title="Riesz representation theorem">Riesz representation</a></li> <li><a href="/wiki/Closed_graph_theorem_(functional_analysis)" title="Closed graph theorem (functional analysis)">Closed graph</a></li> <li><a href="/wiki/Uniform_boundedness_principle" title="Uniform boundedness principle">Uniform boundedness principle</a></li> <li><a href="/wiki/Kakutani_fixed-point_theorem#Infinite-dimensional_generalizations" title="Kakutani fixed-point theorem">Kakutani fixed-point</a></li> <li><a href="/wiki/Krein%E2%80%93Milman_theorem" title="Krein–Milman theorem">Krein–Milman</a></li> <li><a class="mw-selflink selflink">Min–max</a></li> <li><a href="/wiki/Gelfand%E2%80%93Naimark_theorem" title="Gelfand–Naimark theorem">Gelfand–Naimark</a></li> <li><a href="/wiki/Banach%E2%80%93Alaoglu_theorem" title="Banach–Alaoglu theorem">Banach–Alaoglu</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Operators</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Adjoint_operator" class="mw-redirect" title="Adjoint operator">Adjoint</a></li> <li><a href="/wiki/Bounded_operator" title="Bounded operator">Bounded</a></li> <li><a href="/wiki/Compact_operator" title="Compact operator">Compact</a></li> <li><a href="/wiki/Hilbert%E2%80%93Schmidt_operator" title="Hilbert–Schmidt operator">Hilbert–Schmidt</a></li> <li><a href="/wiki/Normal_operator" title="Normal operator">Normal</a></li> <li><a href="/wiki/Nuclear_operator" title="Nuclear operator">Nuclear</a></li> <li><a href="/wiki/Trace_class" title="Trace class">Trace class</a></li> <li><a href="/wiki/Transpose_of_a_linear_map" title="Transpose of a linear map">Transpose</a></li> <li><a href="/wiki/Unbounded_operator" title="Unbounded operator">Unbounded</a></li> <li><a href="/wiki/Unitary_operator" title="Unitary operator">Unitary</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Algebras</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Banach_algebra" title="Banach algebra">Banach algebra</a></li> <li><a href="/wiki/C*-algebra" title="C*-algebra">C*-algebra</a></li> <li><a href="/wiki/Spectrum_of_a_C*-algebra" title="Spectrum of a C*-algebra">Spectrum of a C*-algebra</a></li> <li><a href="/wiki/Operator_algebra" title="Operator algebra">Operator algebra</a></li> <li><a href="/wiki/Group_algebra_of_a_locally_compact_group" title="Group algebra of a locally compact group">Group algebra of a locally compact group</a></li> <li><a href="/wiki/Von_Neumann_algebra" title="Von Neumann algebra">Von Neumann algebra</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Open problems</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Invariant_subspace_problem" title="Invariant subspace problem">Invariant subspace problem</a></li> <li><a href="/wiki/Mahler%27s_conjecture" class="mw-redirect" title="Mahler's conjecture">Mahler's conjecture</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Applications</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Hardy_space" title="Hardy space">Hardy space</a></li> <li><a href="/wiki/Spectral_theory_of_ordinary_differential_equations" title="Spectral theory of ordinary differential equations">Spectral theory of ordinary differential equations</a></li> <li><a href="/wiki/Heat_kernel" title="Heat kernel">Heat kernel</a></li> <li><a href="/wiki/Index_theorem" class="mw-redirect" title="Index theorem">Index theorem</a></li> <li><a href="/wiki/Calculus_of_variations" title="Calculus of variations">Calculus of variations</a></li> <li><a href="/wiki/Functional_calculus" title="Functional calculus">Functional calculus</a></li> <li><a href="/wiki/Integral_operator" title="Integral operator">Integral operator</a></li> <li><a href="/wiki/Jones_polynomial" title="Jones polynomial">Jones polynomial</a></li> <li><a href="/wiki/Topological_quantum_field_theory" title="Topological quantum field theory">Topological quantum field theory</a></li> <li><a href="/wiki/Noncommutative_geometry" title="Noncommutative geometry">Noncommutative geometry</a></li> <li><a href="/wiki/Riemann_hypothesis" title="Riemann hypothesis">Riemann hypothesis</a></li> <li><a href="/wiki/Distribution_(mathematics)" title="Distribution (mathematics)">Distribution</a> (or <a href="/wiki/Generalized_function" title="Generalized function">Generalized functions</a>)</li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Advanced topics</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Approximation_property" title="Approximation property">Approximation property</a></li> <li><a href="/wiki/Balanced_set" title="Balanced set">Balanced set</a></li> <li><a href="/wiki/Choquet_theory" title="Choquet theory">Choquet theory</a></li> <li><a href="/wiki/Weak_topology" title="Weak topology">Weak topology</a></li> <li><a href="/wiki/Banach%E2%80%93Mazur_distance" class="mw-redirect" title="Banach–Mazur distance">Banach–Mazur distance</a></li> <li><a href="/wiki/Tomita%E2%80%93Takesaki_theory" title="Tomita–Takesaki theory">Tomita–Takesaki theory</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div> <ul><li><span class="noviewer" typeof="mw:File"><span title="Category"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/16px-Symbol_category_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/23px-Symbol_category_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/31px-Symbol_category_class.svg.png 2x" data-file-width="180" data-file-height="185" /></span></span> <a href="/wiki/Category:Functional_analysis" title="Category:Functional analysis">Category</a></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"></div><div role="navigation" class="navbox" aria-labelledby="Analysis_in_topological_vector_spaces" style="padding:3px"><table class="nowraplinks hlist mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Analysis_in_topological_vector_spaces" title="Template:Analysis in topological vector spaces"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Analysis_in_topological_vector_spaces" title="Template talk:Analysis in topological vector spaces"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Analysis_in_topological_vector_spaces" title="Special:EditPage/Template:Analysis in topological vector spaces"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Analysis_in_topological_vector_spaces" style="font-size:114%;margin:0 4em"><a href="/wiki/Mathematical_analysis" title="Mathematical analysis">Analysis</a> in <a href="/wiki/Topological_vector_space" title="Topological vector space">topological vector spaces</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Basic concepts</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Abstract_Wiener_space" title="Abstract Wiener space">Abstract Wiener space</a> <ul><li><a href="/wiki/Classical_Wiener_space" title="Classical Wiener space">Classical Wiener space</a></li></ul></li> <li><a href="/wiki/Bochner_space" title="Bochner space">Bochner space</a></li> <li><a href="/wiki/Convex_series" title="Convex series">Convex series</a></li> <li><a href="/wiki/Cylinder_set_measure" title="Cylinder set measure">Cylinder set measure</a></li> <li><a href="/wiki/Infinite-dimensional_vector_function" title="Infinite-dimensional vector function">Infinite-dimensional vector function</a></li> <li><a href="/wiki/Matrix_calculus" title="Matrix calculus">Matrix calculus</a></li> <li><a href="/wiki/Vector_calculus" title="Vector calculus">Vector calculus</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Derivative" title="Derivative">Derivatives</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Differentiable_vector%E2%80%93valued_functions_from_Euclidean_space" title="Differentiable vector–valued functions from Euclidean space">Differentiable vector–valued functions from Euclidean space</a></li> <li><a href="/wiki/Differentiation_in_Fr%C3%A9chet_spaces" title="Differentiation in Fréchet spaces">Differentiation in Fréchet spaces</a></li> <li><a href="/wiki/Fr%C3%A9chet_derivative" title="Fréchet derivative">Fréchet derivative</a> <ul><li><a href="/wiki/Total_derivative" title="Total derivative">Total</a></li></ul></li> <li><a href="/wiki/Functional_derivative" title="Functional derivative">Functional derivative</a></li> <li><a href="/wiki/Gateaux_derivative" title="Gateaux derivative">Gateaux derivative</a> <ul><li><a href="/wiki/Directional_derivative" title="Directional derivative">Directional</a></li></ul></li> <li><a href="/wiki/Generalizations_of_the_derivative" title="Generalizations of the derivative">Generalizations of the derivative</a></li> <li><a href="/wiki/Hadamard_derivative" title="Hadamard derivative">Hadamard derivative</a></li> <li><a href="/wiki/Infinite-dimensional_holomorphy" title="Infinite-dimensional holomorphy">Holomorphic</a></li> <li><a href="/wiki/Quasi-derivative" title="Quasi-derivative">Quasi-derivative</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Measurability</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Besov_measure" title="Besov measure">Besov measure</a></li> <li><a href="/wiki/Cylinder_set_measure" title="Cylinder set measure">Cylinder set measure</a> <ul><li><a href="/wiki/Canonical_Gaussian_cylinder_set_measure" class="mw-redirect" title="Canonical Gaussian cylinder set measure">Canonical Gaussian</a></li> <li><a href="/wiki/Classical_Wiener_measure" class="mw-redirect" title="Classical Wiener measure">Classical Wiener measure</a></li></ul></li> <li><a href="/wiki/Measure_(mathematics)" title="Measure (mathematics)">Measure</a> like <a href="/wiki/Set_function" title="Set function">set functions</a> <ul><li><a href="/wiki/Gaussian_measure#Infinite-dimensional_spaces" title="Gaussian measure">infinite-dimensional Gaussian measure</a></li> <li><a href="/wiki/Projection-valued_measure" title="Projection-valued measure">Projection-valued</a></li> <li><a href="/wiki/Vector_measure" title="Vector measure">Vector</a></li></ul></li> <li><a href="/wiki/Bochner_measurable_function" title="Bochner measurable function">Bochner</a> / <a href="/wiki/Weakly_measurable_function" title="Weakly measurable function">Weakly</a> / <a href="/wiki/Strongly_measurable_function" title="Strongly measurable function">Strongly</a> <a href="/wiki/Measurable_function" title="Measurable function">measurable function</a></li> <li><a href="/wiki/Radonifying_function" title="Radonifying function">Radonifying function</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Integral" title="Integral">Integrals</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Bochner_integral" title="Bochner integral">Bochner</a></li> <li><a href="/wiki/Direct_integral" title="Direct integral">Direct integral</a></li> <li><a href="/wiki/Dunford_integral" class="mw-redirect" title="Dunford integral">Dunford</a></li> <li><a href="/wiki/Pettis_integral" title="Pettis integral">Gelfand–Pettis/Weak</a></li> <li><a href="/wiki/Regulated_integral" title="Regulated integral">Regulated</a></li> <li><a href="/wiki/Paley%E2%80%93Wiener_integral" title="Paley–Wiener integral">Paley–Wiener</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Results</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Cameron%E2%80%93Martin_theorem" title="Cameron–Martin theorem">Cameron–Martin theorem</a></li> <li><a href="/wiki/Inverse_function_theorem" title="Inverse function theorem">Inverse function theorem</a> <ul><li><a href="/wiki/Nash%E2%80%93Moser_theorem" title="Nash–Moser theorem">Nash–Moser theorem</a></li></ul></li> <li><a href="/wiki/Feldman%E2%80%93H%C3%A1jek_theorem" title="Feldman–Hájek theorem">Feldman–Hájek theorem</a></li> <li><a href="/wiki/Infinite-dimensional_Lebesgue_measure" title="Infinite-dimensional Lebesgue measure">No infinite-dimensional Lebesgue measure</a></li> <li><a href="/wiki/Sazonov%27s_theorem" title="Sazonov's theorem">Sazonov's theorem</a></li> <li><a href="/wiki/Structure_theorem_for_Gaussian_measures" title="Structure theorem for Gaussian measures">Structure theorem for Gaussian measures</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Crinkled_arc" title="Crinkled arc">Crinkled arc</a></li> <li><a href="/wiki/Covariance_operator" title="Covariance operator">Covariance operator</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Functional_calculus" title="Functional calculus">Functional calculus</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Borel_functional_calculus" title="Borel functional calculus">Borel functional calculus</a></li> <li><a href="/wiki/Continuous_functional_calculus" title="Continuous functional calculus">Continuous functional calculus</a></li> <li><a href="/wiki/Holomorphic_functional_calculus" title="Holomorphic functional calculus">Holomorphic functional calculus</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Applications</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Banach_manifold" title="Banach manifold">Banach manifold</a> (<a href="/wiki/Banach_bundle" title="Banach bundle">bundle</a>)</li> <li><a href="/wiki/Convenient_vector_space" title="Convenient vector space">Convenient vector space</a></li> <li><a href="/wiki/Choquet_theory" title="Choquet theory">Choquet theory</a></li> <li><a href="/wiki/Fr%C3%A9chet_manifold" title="Fréchet manifold">Fréchet manifold</a></li> <li><a href="/wiki/Hilbert_manifold" title="Hilbert manifold">Hilbert manifold</a></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"></div><div role="navigation" class="navbox" aria-labelledby="Spectral_theory_and_*-algebras" style="padding:3px"><table class="nowraplinks hlist mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Spectral_theory" title="Template:Spectral theory"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Spectral_theory" title="Template talk:Spectral theory"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Spectral_theory" title="Special:EditPage/Template:Spectral theory"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Spectral_theory_and_*-algebras" style="font-size:114%;margin:0 4em"><a href="/wiki/Spectral_theory" title="Spectral theory">Spectral theory</a> and <a href="/wiki/*-algebra" title="*-algebra"><sup>*</sup>-algebras</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Basic concepts</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/*-algebra" title="*-algebra">Involution/*-algebra</a></li> <li><a href="/wiki/Banach_algebra" title="Banach algebra">Banach algebra</a></li> <li><a href="/wiki/Banach_*-algebra" class="mw-redirect" title="Banach *-algebra">B*-algebra</a></li> <li><a href="/wiki/C*-algebra" title="C*-algebra">C*-algebra</a></li> <li><a href="/wiki/Noncommutative_topology" title="Noncommutative topology">Noncommutative topology</a></li> <li><a href="/wiki/Projection-valued_measure" title="Projection-valued measure">Projection-valued measure</a></li> <li><a href="/wiki/Spectrum_(functional_analysis)" title="Spectrum (functional analysis)">Spectrum</a></li> <li><a href="/wiki/Spectrum_of_a_C*-algebra" title="Spectrum of a C*-algebra">Spectrum of a C*-algebra</a></li> <li><a href="/wiki/Spectral_radius" title="Spectral radius">Spectral radius</a></li> <li><a href="/wiki/Operator_space" title="Operator space">Operator space</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Main results</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Gelfand%E2%80%93Mazur_theorem" title="Gelfand–Mazur theorem">Gelfand–Mazur theorem</a></li> <li><a href="/wiki/Gelfand%E2%80%93Naimark_theorem" title="Gelfand–Naimark theorem">Gelfand–Naimark theorem</a></li> <li><a href="/wiki/Gelfand_representation" title="Gelfand representation">Gelfand representation</a></li> <li><a href="/wiki/Polar_decomposition" title="Polar decomposition">Polar decomposition</a></li> <li><a href="/wiki/Singular_value_decomposition" title="Singular value decomposition">Singular value decomposition</a></li> <li><a href="/wiki/Spectral_theorem" title="Spectral theorem">Spectral theorem</a></li> <li><a href="/wiki/Spectral_theory_of_normal_C*-algebras" title="Spectral theory of normal C*-algebras">Spectral theory of normal C*-algebras</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Special Elements/Operators</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Isospectral" title="Isospectral">Isospectral</a></li> <li><a href="/wiki/Normal_element" title="Normal element">Normal</a> <a href="/wiki/Normal_operator" title="Normal operator">operator</a></li> <li><a href="/wiki/Self-adjoint" title="Self-adjoint">Hermitian/Self-adjoint</a> <a href="/wiki/Self-adjoint_operator" title="Self-adjoint operator">operator</a></li> <li><a href="/wiki/Unitary_element" title="Unitary element">Unitary</a> <a href="/wiki/Unitary_operator" title="Unitary operator">operator</a></li> <li><a href="/wiki/Unit_(ring_theory)" title="Unit (ring theory)">Unit</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Spectrum_(functional_analysis)" title="Spectrum (functional analysis)">Spectrum</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Krein%E2%80%93Rutman_theorem" title="Krein–Rutman theorem">Krein–Rutman theorem</a></li> <li><a href="/wiki/Normal_eigenvalue" title="Normal eigenvalue">Normal eigenvalue</a></li> <li><a href="/wiki/Spectrum_of_a_C*-algebra" title="Spectrum of a C*-algebra">Spectrum of a C*-algebra</a></li> <li><a href="/wiki/Spectral_radius" title="Spectral radius">Spectral radius</a></li> <li><a href="/wiki/Spectral_asymmetry" title="Spectral asymmetry">Spectral asymmetry</a></li> <li><a href="/wiki/Spectral_gap" title="Spectral gap">Spectral gap</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Decomposition</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Decomposition_of_spectrum_(functional_analysis)" title="Decomposition of spectrum (functional analysis)">Decomposition of a spectrum</a> <ul><li><a href="/wiki/Continuous_spectrum_(functional_analysis)" class="mw-redirect" title="Continuous spectrum (functional analysis)">Continuous</a></li> <li><a href="/wiki/Point_spectrum" class="mw-redirect" title="Point spectrum">Point</a></li> <li><a href="/wiki/Spectrum_(functional_analysis)#Residual_spectrum" title="Spectrum (functional analysis)">Residual</a></li></ul></li> <li><a href="/wiki/Spectrum_(functional_analysis)#Approximate_point_spectrum" title="Spectrum (functional analysis)">Approximate point</a></li> <li><a href="/wiki/Spectrum_(functional_analysis)#Compression_spectrum" title="Spectrum (functional analysis)">Compression</a></li> <li><a href="/wiki/Direct_integral" title="Direct integral">Direct integral</a></li> <li><a href="/wiki/Discrete_spectrum_(mathematics)" title="Discrete spectrum (mathematics)">Discrete</a></li> <li><a href="/wiki/Spectral_abscissa" title="Spectral abscissa">Spectral abscissa</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Spectral Theorem</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Borel_functional_calculus" title="Borel functional calculus">Borel functional calculus</a></li> <li><a class="mw-selflink selflink">Min-max theorem</a></li> <li><a href="/wiki/Positive_operator-valued_measure" class="mw-redirect" title="Positive operator-valued measure">Positive operator-valued measure</a></li> <li><a href="/wiki/Projection-valued_measure" title="Projection-valued measure">Projection-valued measure</a></li> <li><a href="/wiki/Riesz_projector" title="Riesz projector">Riesz projector</a></li> <li><a href="/wiki/Rigged_Hilbert_space" title="Rigged Hilbert space">Rigged Hilbert space</a></li> <li><a href="/wiki/Spectral_theorem" title="Spectral theorem">Spectral theorem</a></li> <li><a href="/wiki/Spectral_theory_of_compact_operators" title="Spectral theory of compact operators">Spectral theory of compact operators</a></li> <li><a href="/wiki/Spectral_theory_of_normal_C*-algebras" title="Spectral theory of normal C*-algebras">Spectral theory of normal C*-algebras</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Special algebras</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Amenable_Banach_algebra" title="Amenable Banach algebra">Amenable Banach algebra</a></li> <li>With an <a href="/wiki/Approximate_identity" title="Approximate identity">Approximate identity</a></li> <li><a href="/wiki/Banach_function_algebra" title="Banach function algebra">Banach function algebra</a></li> <li><a href="/wiki/Disk_algebra" title="Disk algebra">Disk algebra</a></li> <li><a href="/wiki/Nuclear_C*-algebra" title="Nuclear C*-algebra">Nuclear C*-algebra</a></li> <li><a href="/wiki/Uniform_algebra" title="Uniform algebra">Uniform algebra</a></li> <li><a href="/wiki/Von_Neumann_algebra" title="Von Neumann algebra">Von Neumann algebra</a> <ul><li><a href="/wiki/Tomita%E2%80%93Takesaki_theory" title="Tomita–Takesaki theory">Tomita–Takesaki theory</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Finite-Dimensional</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Alon%E2%80%93Boppana_bound" title="Alon–Boppana bound">Alon–Boppana bound</a></li> <li><a href="/wiki/Bauer%E2%80%93Fike_theorem" title="Bauer–Fike theorem">Bauer–Fike theorem</a></li> <li><a href="/wiki/Numerical_range" title="Numerical range">Numerical range</a></li> <li><a href="/wiki/Schur%E2%80%93Horn_theorem" title="Schur–Horn theorem">Schur–Horn theorem</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Generalizations</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Dirac_spectrum" title="Dirac spectrum">Dirac spectrum</a></li> <li><a href="/wiki/Essential_spectrum" title="Essential spectrum">Essential spectrum</a></li> <li><a href="/wiki/Pseudospectrum" title="Pseudospectrum">Pseudospectrum</a></li> <li><a href="/wiki/Structure_space" class="mw-redirect" title="Structure space">Structure space</a> (<a href="/wiki/Shilov_boundary" title="Shilov boundary">Shilov boundary</a>)</li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Miscellaneous</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Abstract_index_group" class="mw-redirect" title="Abstract index group">Abstract index group</a></li> <li><a href="/wiki/Banach_algebra_cohomology" title="Banach algebra cohomology">Banach algebra cohomology</a></li> <li><a href="/wiki/Cohen%E2%80%93Hewitt_factorization_theorem" title="Cohen–Hewitt factorization theorem">Cohen–Hewitt factorization theorem</a></li> <li><a href="/wiki/Extensions_of_symmetric_operators" title="Extensions of symmetric operators">Extensions of symmetric operators</a></li> <li><a href="/wiki/Fredholm_theory" title="Fredholm theory">Fredholm theory</a></li> <li><a href="/wiki/Limiting_absorption_principle" title="Limiting absorption principle">Limiting absorption principle</a></li> <li><a href="/wiki/Schr%C3%B6der%E2%80%93Bernstein_theorems_for_operator_algebras" title="Schröder–Bernstein theorems for operator algebras">Schröder–Bernstein theorems for operator algebras</a></li> <li><a href="/wiki/Sherman%E2%80%93Takeda_theorem" title="Sherman–Takeda theorem">Sherman–Takeda theorem</a></li> <li><a href="/wiki/Unbounded_operator" title="Unbounded operator">Unbounded operator</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Examples</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Wiener_algebra" title="Wiener algebra">Wiener algebra</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Applications</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Almost_Mathieu_operator" title="Almost Mathieu operator">Almost Mathieu operator</a></li> <li><a href="/wiki/Corona_theorem" title="Corona theorem">Corona theorem</a></li> <li><a href="/wiki/Hearing_the_shape_of_a_drum" title="Hearing the shape of a drum">Hearing the shape of a drum</a> (<a href="/wiki/Dirichlet_eigenvalue" title="Dirichlet eigenvalue">Dirichlet eigenvalue</a>)</li> <li><a href="/wiki/Heat_kernel" title="Heat kernel">Heat kernel</a></li> <li><a href="/wiki/Kuznetsov_trace_formula" title="Kuznetsov trace formula">Kuznetsov trace formula</a></li> <li><a href="/wiki/Lax_pair" title="Lax pair">Lax pair</a></li> <li><a href="/wiki/Proto-value_function" title="Proto-value function">Proto-value function</a></li> <li><a href="/wiki/Ramanujan_graph" title="Ramanujan graph">Ramanujan graph</a></li> <li><a href="/wiki/Rayleigh%E2%80%93Faber%E2%80%93Krahn_inequality" title="Rayleigh–Faber–Krahn inequality">Rayleigh–Faber–Krahn inequality</a></li> <li><a href="/wiki/Spectral_geometry" title="Spectral geometry">Spectral geometry</a></li> <li><a href="/wiki/Spectral_method" title="Spectral method">Spectral method</a></li> <li><a href="/wiki/Spectral_theory_of_ordinary_differential_equations" title="Spectral theory of ordinary differential equations">Spectral theory of ordinary differential equations</a></li> <li><a href="/wiki/Sturm%E2%80%93Liouville_theory" title="Sturm–Liouville theory">Sturm–Liouville theory</a></li> <li><a href="/wiki/Superstrong_approximation" title="Superstrong approximation">Superstrong approximation</a></li> <li><a href="/wiki/Transfer_operator" title="Transfer operator">Transfer operator</a></li> <li><a href="/wiki/Transform_theory" title="Transform theory">Transform theory</a></li> <li><a href="/wiki/Weyl_law" title="Weyl law">Weyl law</a></li> <li><a href="/wiki/Wiener%E2%80%93Khinchin_theorem" title="Wiener–Khinchin theorem">Wiener–Khinchin theorem</a></li></ul> </div></td></tr></tbody></table></div>    </div><noscript><img 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Min-max theorem - Wikipedia