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href="/nlab/show/stable+homotopy+theory">stable</a>, <a class="existingWikiWord" href="/nlab/show/equivariant+homotopy+theory">equivariant</a>, <a class="existingWikiWord" href="/nlab/show/rational+homotopy+theory">rational</a>, <a class="existingWikiWord" href="/nlab/show/p-adic+homotopy+theory">p-adic</a>, <a class="existingWikiWord" href="/nlab/show/proper+homotopy+theory">proper</a>, <a class="existingWikiWord" href="/nlab/show/geometric+homotopy+theory">geometric</a>, <a class="existingWikiWord" href="/nlab/show/cohesive+homotopy+theory">cohesive</a>, <a class="existingWikiWord" href="/nlab/show/directed+homotopy+theory">directed</a>…</p> <p>models: <a class="existingWikiWord" href="/nlab/show/topological+homotopy+theory">topological</a>, <a class="existingWikiWord" href="/nlab/show/simplicial+homotopy+theory">simplicial</a>, <a class="existingWikiWord" href="/nlab/show/localic+homotopy+theory">localic</a>, …</p> <p>see also <strong><a class="existingWikiWord" href="/nlab/show/algebraic+topology">algebraic topology</a></strong></p> <p><strong>Introductions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Introduction+to+Topology+--+2">Introduction to Basic Homotopy Theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Introduction+to+Homotopy+Theory">Introduction to Abstract Homotopy Theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+homotopy+types">geometry of physics – homotopy types</a></p> </li> </ul> <p><strong>Definitions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy">homotopy</a>, <a class="existingWikiWord" href="/nlab/show/higher+homotopy">higher homotopy</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+type">homotopy type</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Pi-algebra">Pi-algebra</a>, <a class="existingWikiWord" href="/nlab/show/spherical+object+and+Pi%28A%29-algebra">spherical object and Pi(A)-algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+coherent+category+theory">homotopy coherent category theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopical+category">homotopical category</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+category">model category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/category+of+fibrant+objects">category of fibrant objects</a>, <a class="existingWikiWord" href="/nlab/show/cofibration+category">cofibration category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Waldhausen+category">Waldhausen category</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+category">homotopy category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Ho%28Top%29">Ho(Top)</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/homotopy+category+of+an+%28%E2%88%9E%2C1%29-category">homotopy category of an (∞,1)-category</a></li> </ul> </li> </ul> <p><strong>Paths and cylinders</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/left+homotopy">left homotopy</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cylinder+object">cylinder object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mapping+cone">mapping cone</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/right+homotopy">right homotopy</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/path+object">path object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mapping+cocone">mapping cocone</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/generalized+universal+bundle">universal bundle</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/interval+object">interval object</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+localization">homotopy localization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/infinitesimal+interval+object">infinitesimal interval object</a></p> </li> </ul> </li> </ul> <p><strong>Homotopy groups</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+group">homotopy group</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+group">fundamental group</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/fundamental+group+of+a+topos">fundamental group of a topos</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Brown-Grossman+homotopy+group">Brown-Grossman homotopy group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/categorical+homotopy+groups+in+an+%28%E2%88%9E%2C1%29-topos">categorical homotopy groups in an (∞,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometric+homotopy+groups+in+an+%28%E2%88%9E%2C1%29-topos">geometric homotopy groups in an (∞,1)-topos</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid">fundamental ∞-groupoid</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+groupoid">fundamental groupoid</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/path+groupoid">path groupoid</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid+in+a+locally+%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid+of+a+locally+%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">fundamental ∞-groupoid of a locally ∞-connected (∞,1)-topos</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+%28%E2%88%9E%2C1%29-category">fundamental (∞,1)-category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/fundamental+category">fundamental category</a></li> </ul> </li> </ul> <p><strong>Basic facts</strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/fundamental+group+of+the+circle+is+the+integers">fundamental group of the circle is the integers</a></li> </ul> <p><strong>Theorems</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+theorem+of+covering+spaces">fundamental theorem of covering spaces</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Freudenthal+suspension+theorem">Freudenthal suspension theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Blakers-Massey+theorem">Blakers-Massey theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+homotopy+van+Kampen+theorem">higher homotopy van Kampen theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/nerve+theorem">nerve theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Whitehead%27s+theorem">Whitehead's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hurewicz+theorem">Hurewicz theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Galois+theory">Galois theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+hypothesis">homotopy hypothesis</a>-theorem</p> </li> </ul> </div></div> <h4 id="algebra">Algebra</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/higher+algebra">higher algebra</a></strong></p> <p><a class="existingWikiWord" href="/nlab/show/universal+algebra">universal algebra</a></p> <h2 id="algebraic_theories">Algebraic theories</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/algebraic+theory">algebraic theory</a> / <a class="existingWikiWord" href="/nlab/show/2-algebraic+theory">2-algebraic theory</a> / <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-algebraic+theory">(∞,1)-algebraic theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monad">monad</a> / <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-monad">(∞,1)-monad</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/operad">operad</a> / <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-operad">(∞,1)-operad</a></p> </li> </ul> <h2 id="algebras_and_modules">Algebras and modules</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/algebra+over+a+monad">algebra over a monad</a></p> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-algebra+over+an+%28%E2%88%9E%2C1%29-monad">∞-algebra over an (∞,1)-monad</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/algebra+over+an+algebraic+theory">algebra over an algebraic theory</a></p> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-algebra+over+an+%28%E2%88%9E%2C1%29-algebraic+theory">∞-algebra over an (∞,1)-algebraic theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/algebra+over+an+operad">algebra over an operad</a></p> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-algebra+over+an+%28%E2%88%9E%2C1%29-operad">∞-algebra over an (∞,1)-operad</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/action">action</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-action">∞-action</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/representation">representation</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-representation">∞-representation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/module">module</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-module">∞-module</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/associated+bundle">associated bundle</a>, <a class="existingWikiWord" href="/nlab/show/associated+%E2%88%9E-bundle">associated ∞-bundle</a></p> </li> </ul> <h2 id="higher_algebras">Higher algebras</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/monoidal+%28%E2%88%9E%2C1%29-category">monoidal (∞,1)-category</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+%28%E2%88%9E%2C1%29-category">symmetric monoidal (∞,1)-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monoid+in+an+%28%E2%88%9E%2C1%29-category">monoid in an (∞,1)-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/commutative+monoid+in+an+%28%E2%88%9E%2C1%29-category">commutative monoid in an (∞,1)-category</a></p> </li> </ul> </li> <li> <p>symmetric monoidal (∞,1)-category of spectra</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/smash+product+of+spectra">smash product of spectra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+smash+product+of+spectra">symmetric monoidal smash product of spectra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/ring+spectrum">ring spectrum</a>, <a class="existingWikiWord" href="/nlab/show/module+spectrum">module spectrum</a>, <a class="existingWikiWord" href="/nlab/show/algebra+spectrum">algebra spectrum</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/A-%E2%88%9E+algebra">A-∞ algebra</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/A-%E2%88%9E+ring">A-∞ ring</a>, <a class="existingWikiWord" href="/nlab/show/A-%E2%88%9E+space">A-∞ space</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/C-%E2%88%9E+algebra">C-∞ algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/E-%E2%88%9E+ring">E-∞ ring</a>, <a class="existingWikiWord" href="/nlab/show/E-%E2%88%9E+algebra">E-∞ algebra</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-module">∞-module</a>, <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-module+bundle">(∞,1)-module bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/multiplicative+cohomology+theory">multiplicative cohomology theory</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/L-%E2%88%9E+algebra">L-∞ algebra</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/deformation+theory">deformation theory</a></li> </ul> </li> </ul> <h2 id="model_category_presentations">Model category presentations</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+T-algebras">model structure on simplicial T-algebras</a> / <a class="existingWikiWord" href="/nlab/show/homotopy+T-algebra">homotopy T-algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+operads">model structure on operads</a></p> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+algebras+over+an+operad">model structure on algebras over an operad</a></p> </li> </ul> <h2 id="geometry_on_formal_duals_of_algebras">Geometry on formal duals of algebras</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Isbell+duality">Isbell duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/derived+geometry">derived geometry</a></p> </li> </ul> <h2 id="theorems">Theorems</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Deligne+conjecture">Deligne conjecture</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/delooping+hypothesis">delooping hypothesis</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monoidal+Dold-Kan+correspondence">monoidal Dold-Kan correspondence</a></p> </li> </ul> <div> <p> <a href="/nlab/edit/higher+algebra+-+contents">Edit this sidebar</a> </p> </div></div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#definition'>Definition</a></li> <ul> <li><a href='#preliminaries_on_exterior_algebra'>Preliminaries on exterior algebra</a></li> <li><a href='#determinant_of_a_matrix'>Determinant of a matrix</a></li> </ul> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#cramers_rule'>Cramer’s rule</a></li> <li><a href='#characteristic_polynomial_and_cayleyhamilton_theorem'>Characteristic polynomial and Cayley-Hamilton theorem</a></li> <li><a href='#over_the_real_numbers_volume_and_orientation'>Over the real numbers: volume and orientation</a></li> <li><a href='#AsAPolynomialInTracesofPowers'>As a polynomial in traces of powers</a></li> </ul> <li><a href='#in_terms_of_berezinian_integrals'>In terms of Berezinian integrals</a></li> <li><a href='#related_entries'>Related entries</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>The determinant is the (essentially unique) universal alternating <a class="existingWikiWord" href="/nlab/show/multilinear+map">multilinear map</a>.</p> <h2 id="definition">Definition</h2> <h3 id="preliminaries_on_exterior_algebra">Preliminaries on exterior algebra</h3> <p>Let <a class="existingWikiWord" href="/nlab/show/Vect">Vect</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mrow></mrow> <mi>k</mi></msub></mrow><annotation encoding="application/x-tex">{}_k</annotation></semantics></math> be the <a class="existingWikiWord" href="/nlab/show/category">category</a> of <a class="existingWikiWord" href="/nlab/show/vector+space">vector spaces</a> over a <a class="existingWikiWord" href="/nlab/show/field">field</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>, and assume for the moment that the <a class="existingWikiWord" href="/nlab/show/characteristic">characteristic</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>char</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mo>≠</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">char(k) \neq 2</annotation></semantics></math>. For each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>j</mi><mo>≥</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">j \geq 0</annotation></semantics></math>, let</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>sgn</mi> <mi>j</mi></msub><mo lspace="verythinmathspace">:</mo><msub><mi>S</mi> <mi>j</mi></msub><mo>→</mo><mi>hom</mi><mo stretchy="false">(</mo><mi>k</mi><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">sgn_j \colon S_j \to \hom(k, k)</annotation></semantics></math></div> <p>be the 1-dimensional <a class="existingWikiWord" href="/nlab/show/sign+representation">sign representation</a> on the <a class="existingWikiWord" href="/nlab/show/symmetric+group">symmetric group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>S</mi> <mi>j</mi></msub></mrow><annotation encoding="application/x-tex">S_j</annotation></semantics></math>, taking each transposition <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>i</mi><mi>j</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(i j)</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><mo>∈</mo><msup><mi>k</mi> <mo>×</mo></msup></mrow><annotation encoding="application/x-tex">-1 \in k^\times</annotation></semantics></math>. We may linearly extend the sign action of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>S</mi> <mi>j</mi></msub></mrow><annotation encoding="application/x-tex">S_j</annotation></semantics></math>, so that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>sgn</mi></mrow><annotation encoding="application/x-tex">sgn</annotation></semantics></math> names a (right) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><msub><mi>S</mi> <mi>j</mi></msub></mrow><annotation encoding="application/x-tex">k S_j</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/module">module</a> with underlying <a class="existingWikiWord" href="/nlab/show/vector+space">vector space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>. At the same time, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>S</mi> <mi>j</mi></msub></mrow><annotation encoding="application/x-tex">S_j</annotation></semantics></math> acts on the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>j</mi> <mi>th</mi></msup></mrow><annotation encoding="application/x-tex">j^{th}</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/tensor+product">tensor product</a> of a vector space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math> by permuting tensor factors, giving a left <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><msub><mi>S</mi> <mi>j</mi></msub></mrow><annotation encoding="application/x-tex">k S_j</annotation></semantics></math>-module structure on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>V</mi> <mrow><mo>⊗</mo><mi>j</mi></mrow></msup></mrow><annotation encoding="application/x-tex">V^{\otimes j}</annotation></semantics></math>. We define the <a class="existingWikiWord" href="/nlab/show/Schur+functor">Schur functor</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>Λ</mi> <mi>j</mi></msup><mo lspace="verythinmathspace">:</mo><msub><mi>Vect</mi> <mi>k</mi></msub><mo>→</mo><msub><mi>Vect</mi> <mi>k</mi></msub></mrow><annotation encoding="application/x-tex">\Lambda^j \colon Vect_k \to Vect_k</annotation></semantics></math></div> <p>by the formula</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>Λ</mi> <mi>j</mi></msup><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>sgn</mi> <mi>j</mi></msub><msub><mo>⊗</mo> <mrow><mi>k</mi><msub><mi>S</mi> <mi>j</mi></msub></mrow></msub><msup><mi>V</mi> <mrow><mo>⊗</mo><mi>j</mi></mrow></msup><mo>.</mo></mrow><annotation encoding="application/x-tex">\Lambda^j(V) = sgn_j \otimes_{k S_j} V^{\otimes j}.</annotation></semantics></math></div> <p>It is called the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>j</mi> <mi>th</mi></msup></mrow><annotation encoding="application/x-tex">j^{th}</annotation></semantics></math> <strong>alternating power</strong> (of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math>).</p> <p>Another point of view on the alternating power is via <a class="existingWikiWord" href="/nlab/show/superalgebra">superalgebra</a>. For any <a class="existingWikiWord" href="/nlab/show/cosmos">cosmos</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>V</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{V}</annotation></semantics></math> let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>CMon</mi><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>V</mi></mstyle><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">CMon(\mathbf{V})</annotation></semantics></math> be the category of <a class="existingWikiWord" href="/nlab/show/commutative+monoid">commutative monoid</a> objects in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>V</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{V}</annotation></semantics></math>. The forgetful functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>CMon</mi><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>V</mi></mstyle><mo stretchy="false">)</mo><mo>→</mo><mstyle mathvariant="bold"><mi>V</mi></mstyle></mrow><annotation encoding="application/x-tex">CMon(\mathbf{V}) \to \mathbf{V}</annotation></semantics></math> has a <a class="existingWikiWord" href="/nlab/show/left+adjoint">left adjoint</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>exp</mi><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">)</mo><mo>=</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>n</mi><mo>≥</mo><mn>0</mn></mrow></munder><msup><mi>V</mi> <mrow><mo>⊗</mo><mi>n</mi></mrow></msup><mo stretchy="false">/</mo><msub><mi>S</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">\exp(V) = \sum_{n \geq 0} V^{\otimes n}/S_n</annotation></semantics></math></div> <p>whose values are naturally regarded as graded by degree <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>.</p> <p>This applies in particular to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>V</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{V}</annotation></semantics></math> the category of <a class="existingWikiWord" href="/nlab/show/supervector+spaces">supervector spaces</a>; if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math> is a supervector space concentrated in odd degree, say with component <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>V</mi> <mi>odd</mi></msub></mrow><annotation encoding="application/x-tex">V_{odd}</annotation></semantics></math>, then each symmetry <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>σ</mi><mo>:</mo><mi>V</mi><mo>⊗</mo><mi>V</mi><mo>→</mo><mi>V</mi><mo>⊗</mo><mi>V</mi></mrow><annotation encoding="application/x-tex">\sigma: V \otimes V \to V \otimes V</annotation></semantics></math> maps <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>v</mi><mo>⊗</mo><mi>w</mi><mo>↦</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>w</mi><mo>⊗</mo><mi>v</mi></mrow><annotation encoding="application/x-tex">v \otimes w \mapsto -w \otimes v</annotation></semantics></math> for elements <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>v</mi><mo>,</mo><mi>w</mi><mo>∈</mo><msub><mi>V</mi> <mi>odd</mi></msub></mrow><annotation encoding="application/x-tex">v, w \in V_{odd}</annotation></semantics></math>. It follows that the graded component <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>exp</mi><mo stretchy="false">(</mo><mi>V</mi><msub><mo stretchy="false">)</mo> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">\exp(V)_n</annotation></semantics></math> is concentrated in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>parity</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">parity(n)</annotation></semantics></math> degree, with component <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Λ</mi> <mi>n</mi></msup><mo stretchy="false">(</mo><msub><mi>V</mi> <mi>odd</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Lambda^n(V_{odd})</annotation></semantics></math>.</p> <div class="num_prop"> <h6 id="proposition">Proposition</h6> <p>There is a canonical natural isomorphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Λ</mi> <mi>n</mi></msup><mo stretchy="false">(</mo><mi>V</mi><mo>⊕</mo><mi>W</mi><mo stretchy="false">)</mo><mo>≅</mo><msub><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>j</mi><mo>+</mo><mi>k</mi><mo>=</mo><mi>n</mi></mrow></msub><msup><mi>Λ</mi> <mi>j</mi></msup><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">)</mo><mo>⊗</mo><msup><mi>Λ</mi> <mi>k</mi></msup><mo stretchy="false">(</mo><mi>W</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Lambda^n(V \oplus W) \cong \sum_{j + k = n} \Lambda^j(V) \otimes \Lambda^k(W)</annotation></semantics></math>.</p> </div> <div class="proof"> <h6 id="proof">Proof</h6> <p>Again take <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>V</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{V}</annotation></semantics></math> to be the category of supervector spaces. Since the left adjoint <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>exp</mi><mo>:</mo><mstyle mathvariant="bold"><mi>V</mi></mstyle><mo>→</mo><mi>CMon</mi><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>V</mi></mstyle><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\exp: \mathbf{V} \to CMon(\mathbf{V})</annotation></semantics></math> preserves <a class="existingWikiWord" href="/nlab/show/coproducts">coproducts</a> and since the tensor product <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊗</mo></mrow><annotation encoding="application/x-tex">\otimes</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>V</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{V}</annotation></semantics></math> provides the coproduct for commutative monoid objects, we have a natural isomorphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>exp</mi><mo stretchy="false">(</mo><mi>V</mi><mo>⊕</mo><mi>W</mi><mo stretchy="false">)</mo><mo>≅</mo><mi>exp</mi><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">)</mo><mo>⊗</mo><mi>exp</mi><mo stretchy="false">(</mo><mi>W</mi><mo stretchy="false">)</mo><mo>.</mo></mrow><annotation encoding="application/x-tex">\exp(V \oplus W) \cong \exp(V) \otimes \exp(W).</annotation></semantics></math></div> <p>Examining the grade <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math> component <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>exp</mi><mo stretchy="false">(</mo><mi>V</mi><mo>⊕</mo><mi>W</mi><msub><mo stretchy="false">)</mo> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">\exp(V \oplus W)_n</annotation></semantics></math>, this leads to an identification</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>exp</mi><mo stretchy="false">(</mo><mi>V</mi><mo>⊕</mo><mi>W</mi><msub><mo stretchy="false">)</mo> <mi>n</mi></msub><mo>=</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>j</mi><mo>+</mo><mi>k</mi><mo>=</mo><mi>n</mi></mrow></munder><mi>exp</mi><mo stretchy="false">(</mo><mi>V</mi><msub><mo stretchy="false">)</mo> <mi>j</mi></msub><mo>⊗</mo><mi>exp</mi><mo stretchy="false">(</mo><mi>W</mi><msub><mo stretchy="false">)</mo> <mi>k</mi></msub><mo>.</mo></mrow><annotation encoding="application/x-tex">\exp(V \oplus W)_n = \sum_{j + k = n} \exp(V)_j \otimes \exp(W)_k.</annotation></semantics></math></div> <p>and now the result follows by considering the case where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi><mo>,</mo><mi>W</mi></mrow><annotation encoding="application/x-tex">V, W</annotation></semantics></math> are concentrated in odd degree.</p> </div> <div class="num_cor"> <h6 id="corollary">Corollary</h6> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math> is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/dimension">dimensional</a>, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Λ</mi> <mi>j</mi></msup><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Lambda^j(V)</annotation></semantics></math> has dimension <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mrow><mo>(</mo><mfrac linethickness="0"><mi>n</mi><mi>j</mi></mfrac><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex">\binom{n}{j}</annotation></semantics></math>. In particular, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Λ</mi> <mi>n</mi></msup><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Lambda^n(V)</annotation></semantics></math> is 1-dimensional.</p> </div> <div class="proof"> <h6 id="proof_2">Proof</h6> <p>By induction on dimension. If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>dim</mi><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">)</mo><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">\dim(V) = 1</annotation></semantics></math>, we have that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Λ</mi> <mn>0</mn></msup><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Lambda^0(V)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Λ</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Lambda^1(V)</annotation></semantics></math> are <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>1</mn></mrow><annotation encoding="application/x-tex">1</annotation></semantics></math>-dimensional, and clearly <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Λ</mi> <mi>n</mi></msup><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\Lambda^n(V) = 0</annotation></semantics></math> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>≥</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">n \geq 2</annotation></semantics></math>, at least when <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>char</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mo>≠</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">char(k) \neq 2</annotation></semantics></math>.</p> <p>We then infer</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msup><mi>Λ</mi> <mi>j</mi></msup><mo stretchy="false">(</mo><mi>V</mi><mo>⊕</mo><mi>k</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>≅</mo></mtd> <mtd><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>p</mi><mo>+</mo><mi>q</mi><mo>=</mo><mi>j</mi></mrow></munder><msup><mi>Λ</mi> <mi>p</mi></msup><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">)</mo><mo>⊗</mo><msup><mi>Λ</mi> <mi>q</mi></msup><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≅</mo></mtd> <mtd><msup><mi>Λ</mi> <mi>j</mi></msup><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">)</mo><mo>⊕</mo><msup><mi>Λ</mi> <mrow><mi>j</mi><mo>−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex">\array{ \Lambda^j(V \oplus k) & \cong & \sum_{p + q = j} \Lambda^p(V) \otimes \Lambda^q(k) \\ & \cong & \Lambda^j(V) \oplus \Lambda^{j-1}(V) } </annotation></semantics></math></div> <p>where the dimensions satisfy the same recurrence relation as for <a class="existingWikiWord" href="/nlab/show/binomial+coefficients">binomial coefficients</a>: <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mrow><mo>(</mo><mfrac linethickness="0"><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow><mi>j</mi></mfrac><mo>)</mo></mrow><mo>=</mo><mrow><mo>(</mo><mfrac linethickness="0"><mi>n</mi><mi>j</mi></mfrac><mo>)</mo></mrow><mo>+</mo><mrow><mo>(</mo><mfrac linethickness="0"><mi>n</mi><mrow><mi>j</mi><mo>−</mo><mn>1</mn></mrow></mfrac><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex">\binom{n+1}{j} = \binom{n}{j} + \binom{n}{j-1}</annotation></semantics></math>.</p> </div> <p>More concretely: if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>e</mi> <mn>1</mn></msub><mo>,</mo><mi>…</mi><mo>,</mo><msub><mi>e</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">e_1, \ldots, e_n</annotation></semantics></math> is a basis for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math>, then expressions of the form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>e</mi> <mrow><msub><mi>n</mi> <mn>1</mn></msub></mrow></msub><mo>⊗</mo><mi>…</mi><mo>⊗</mo><msub><mi>e</mi> <mrow><msub><mi>n</mi> <mi>j</mi></msub></mrow></msub></mrow><annotation encoding="application/x-tex">e_{n_1} \otimes \ldots \otimes e_{n_j}</annotation></semantics></math> form a <a class="existingWikiWord" href="/nlab/show/basis+of+a+vector+space">basis</a> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>V</mi> <mrow><mo>⊗</mo><mi>j</mi></mrow></msup></mrow><annotation encoding="application/x-tex">V^{\otimes j}</annotation></semantics></math>. Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>e</mi> <mrow><msub><mi>n</mi> <mn>1</mn></msub></mrow></msub><mo>∧</mo><mi>…</mi><mo>∧</mo><msub><mi>e</mi> <mrow><msub><mi>n</mi> <mi>j</mi></msub></mrow></msub></mrow><annotation encoding="application/x-tex">e_{n_1} \wedge \ldots \wedge e_{n_j}</annotation></semantics></math> denote the <a class="existingWikiWord" href="/nlab/show/image">image</a> of this element under the <a class="existingWikiWord" href="/nlab/show/quotient">quotient</a> map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>V</mi> <mrow><mo>⊗</mo><mi>j</mi></mrow></msup><mo>→</mo><msup><mi>Λ</mi> <mi>j</mi></msup><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">V^{\otimes j} \to \Lambda^j(V)</annotation></semantics></math>. We have</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>e</mi> <mrow><msub><mi>n</mi> <mn>1</mn></msub></mrow></msub><mo>∧</mo><mi>…</mi><mo>∧</mo><msub><mi>e</mi> <mrow><msub><mi>n</mi> <mi>i</mi></msub></mrow></msub><mo>∧</mo><msub><mi>e</mi> <mrow><msub><mi>n</mi> <mrow><mi>i</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow></msub><mo>∧</mo><mi>…</mi><mo>∧</mo><msub><mi>e</mi> <mrow><msub><mi>n</mi> <mi>j</mi></msub></mrow></msub><mo>=</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msub><mi>e</mi> <mrow><msub><mi>n</mi> <mn>1</mn></msub></mrow></msub><mo>∧</mo><mi>…</mi><mo>∧</mo><msub><mi>e</mi> <mrow><msub><mi>n</mi> <mrow><mi>i</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow></msub><mo>∧</mo><msub><mi>e</mi> <mrow><msub><mi>n</mi> <mi>i</mi></msub></mrow></msub><mo>∧</mo><mi>…</mi><mo>∧</mo><msub><mi>e</mi> <mrow><msub><mi>n</mi> <mi>j</mi></msub></mrow></msub></mrow><annotation encoding="application/x-tex">e_{n_1} \wedge \ldots \wedge e_{n_i} \wedge e_{n_{i+1}} \wedge \ldots \wedge e_{n_j} = -e_{n_1} \wedge \ldots \wedge e_{n_{i+1}} \wedge e_{n_i} \wedge \ldots \wedge e_{n_j}</annotation></semantics></math></div> <p>(consider the transposition in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>S</mi> <mi>j</mi></msub></mrow><annotation encoding="application/x-tex">S_j</annotation></semantics></math> which swaps <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo>+</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">i+1</annotation></semantics></math>) and so we may take only such expressions on the left where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>n</mi> <mn>1</mn></msub><mo><</mo><mi>…</mi><mo><</mo><msub><mi>n</mi> <mi>j</mi></msub></mrow><annotation encoding="application/x-tex">n_1 \lt \ldots \lt n_j</annotation></semantics></math> as forming a spanning set for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Λ</mi> <mi>j</mi></msup><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Lambda^j(V)</annotation></semantics></math>, and indeed these form a basis. The number of such expressions is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mrow><mo>(</mo><mfrac linethickness="0"><mi>n</mi><mi>j</mi></mfrac><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex">\binom{n}{j}</annotation></semantics></math>.</p> <div class="num_remark"> <h6 id="remark">Remark</h6> <p>In the case where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>char</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mo>=</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">char(k) = 2</annotation></semantics></math>, the same development may be carried out by simply decreeing that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>e</mi> <mrow><msub><mi>n</mi> <mn>1</mn></msub></mrow></msub><mo>∧</mo><mi>…</mi><mo>∧</mo><msub><mi>e</mi> <mrow><msub><mi>n</mi> <mi>j</mi></msub></mrow></msub><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">e_{n_1} \wedge \ldots \wedge e_{n_j} = 0</annotation></semantics></math> whenever <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>n</mi> <mi>i</mi></msub><mo>=</mo><msub><mi>n</mi> <mrow><mi>i</mi><mo>′</mo></mrow></msub></mrow><annotation encoding="application/x-tex">n_i = n_{i'}</annotation></semantics></math> for some pair of distinct indices <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">i'</annotation></semantics></math>.</p> </div> <p>Now let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math> be an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-dimensional space, and let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><mi>V</mi><mo>→</mo><mi>V</mi></mrow><annotation encoding="application/x-tex">f \colon V \to V</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/linear+map">linear map</a>. By the corollary, the map</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>Λ</mi> <mi>n</mi></msup><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo lspace="verythinmathspace">:</mo><msup><mi>Λ</mi> <mi>n</mi></msup><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">)</mo><mo>→</mo><msup><mi>Λ</mi> <mi>n</mi></msup><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">)</mo><mo>,</mo></mrow><annotation encoding="application/x-tex">\Lambda^n(f) \colon \Lambda^n(V) \to \Lambda^n(V),</annotation></semantics></math></div> <p>being an <a class="existingWikiWord" href="/nlab/show/endomorphism">endomorphism</a> on a 1-dimensional space, is given by multiplying by a scalar <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo>∈</mo><mi>k</mi></mrow><annotation encoding="application/x-tex">D(f) \in k</annotation></semantics></math>. It is manifestly <a class="existingWikiWord" href="/nlab/show/functor">functorial</a> since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Λ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\Lambda^n</annotation></semantics></math> is, i.e., <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi><mo stretchy="false">(</mo><mi>f</mi><mi>g</mi><mo stretchy="false">)</mo><mo>=</mo><mi>D</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mi>D</mi><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">D(f g) = D(f) D(g)</annotation></semantics></math>. The quantity <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">D(f)</annotation></semantics></math> is called the <strong>determinant</strong> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math>.</p> <h3 id="determinant_of_a_matrix">Determinant of a matrix</h3> <p>We see then that if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math> is of dimension <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>,</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>det</mi><mo lspace="verythinmathspace">:</mo><mi>End</mi><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">)</mo><mo>→</mo><mi>k</mi></mrow><annotation encoding="application/x-tex">\det \colon End(V) \to k</annotation></semantics></math></div> <p>is a <a class="existingWikiWord" href="/nlab/show/homomorphism">homomorphism</a> of multiplicative <a class="existingWikiWord" href="/nlab/show/monoids">monoids</a>; by commutativity of multiplication in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>, we infer that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>det</mi><mo stretchy="false">(</mo><mi>U</mi><mi>A</mi><msup><mi>U</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">)</mo><mo>=</mo><mi>det</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\det(U A U^{-1}) = \det(A)</annotation></semantics></math></div> <p>for each <a class="existingWikiWord" href="/nlab/show/inverse">invertible</a> <a class="existingWikiWord" href="/nlab/show/linear+map">linear map</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo>∈</mo><mi>GL</mi><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">U \in GL(V)</annotation></semantics></math>.</p> <p>If we choose a <a class="existingWikiWord" href="/nlab/show/basis+of+a+vector+space">basis</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math> so that we have an identification <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>End</mi><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">)</mo><mo>≅</mo><msub><mi>Mat</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">End(V) \cong Mat_n(k)</annotation></semantics></math>, then the determinant gives a <a class="existingWikiWord" href="/nlab/show/function">function</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>det</mi><mo lspace="verythinmathspace">:</mo><msub><mi>Mat</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mo>→</mo><mi>k</mi></mrow><annotation encoding="application/x-tex">\det \colon Mat_n(k) \to k</annotation></semantics></math></div> <p>or by restriction to invertible matrices with invertible determinants a function</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>det</mi><mo lspace="verythinmathspace">:</mo><msub><mi>GL</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mo>→</mo><msup><mi>k</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">\det\colon GL_n(k) \to k^*</annotation></semantics></math></div> <p>that takes products of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>×</mo><mi>n</mi></mrow><annotation encoding="application/x-tex">n \times n</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/matrices">matrices</a> to products in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>. The determinant however is of course independent of choice of basis, since any two choices are related by a change-of-basis matrix <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi></mrow><annotation encoding="application/x-tex">U</annotation></semantics></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> and its transform <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mi>A</mi><msup><mi>U</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">U A U^{-1}</annotation></semantics></math> have the same determinant. The above map is furthermore natural in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>, hence is a <a class="existingWikiWord" href="/nlab/show/natural+transformation">natural transformation</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>det</mi><mo lspace="verythinmathspace">:</mo><msub><mi>GL</mi> <mi>n</mi></msub><mo>→</mo><msup><mo lspace="verythinmathspace" rspace="0em">−</mo> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">det\colon GL_n\rightarrow -^*</annotation></semantics></math> from the <a class="existingWikiWord" href="/nlab/show/general+linear+group">general linear group</a> to the <a class="existingWikiWord" href="/nlab/show/group+of+units">group of units</a> of a <a class="existingWikiWord" href="/nlab/show/field">field</a> (or more generally a <a class="existingWikiWord" href="/nlab/show/ring">ring</a>), which are both functors from <a class="existingWikiWord" href="/nlab/show/Field">Field</a> (or more generally <a class="existingWikiWord" href="/nlab/show/Ring">Ring</a>) to <a class="existingWikiWord" href="/nlab/show/Grp">Grp</a>.</p> <p>By following the definitions above, we can give an explicit formula:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>det</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>σ</mi><mo>∈</mo><msub><mi>S</mi> <mi>n</mi></msub></mrow></munder><mi>sgn</mi><mo stretchy="false">(</mo><mi>σ</mi><mo stretchy="false">)</mo><munderover><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow> <mi>n</mi></munderover><msub><mi>a</mi> <mrow><mi>i</mi><mi>σ</mi><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo></mrow></msub><mspace width="thickmathspace"></mspace></mrow><annotation encoding="application/x-tex"> \det(A) \;=\; \sum_{\sigma \in S_n} sgn(\sigma) \prod_{i = 1}^n a_{i \sigma(i)} \; </annotation></semantics></math></div> <p>This may equivalently be written using the <a class="existingWikiWord" href="/nlab/show/Levi-Civita+symbol">Levi-Civita symbol</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϵ</mi></mrow><annotation encoding="application/x-tex">\epsilon</annotation></semantics></math> and the <a class="existingWikiWord" href="/nlab/show/Einstein+summation+convention">Einstein summation convention</a> as</p> <div class="maruku-equation" id="eq:DeterminantInTermsOfLCSymbol"><span class="maruku-eq-number">(1)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>det</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><msub><mi>a</mi> <mrow><mn>1</mn><msub><mi>j</mi> <mn>1</mn></msub></mrow></msub><msub><mi>a</mi> <mrow><mn>2</mn><msub><mi>j</mi> <mn>2</mn></msub></mrow></msub><mi>⋯</mi><msub><mi>a</mi> <mrow><mi>n</mi><msub><mi>j</mi> <mi>n</mi></msub></mrow></msub><mspace width="thinmathspace"></mspace><msup><mi>ϵ</mi> <mrow><msub><mi>j</mi> <mn>1</mn></msub><msub><mi>j</mi> <mn>2</mn></msub><mi>⋯</mi><msub><mi>j</mi> <mi>n</mi></msub></mrow></msup></mrow><annotation encoding="application/x-tex"> \det(A) \;=\; a_{1 j_1} a_{2 j_2} \cdots a_{n j_n} \, \epsilon^{j_1 j_2 \cdots j_n} </annotation></semantics></math></div> <p>which in turn may be re-written more symmetrically as</p> <div class="maruku-equation" id="eq:DeterminantInTermsOfLCSymbols"><span class="maruku-eq-number">(2)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>det</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><mfrac><mn>1</mn><mrow><mi>n</mi><mo>!</mo></mrow></mfrac><msup><mi>ϵ</mi> <mrow><msub><mi>i</mi> <mn>1</mn></msub><msub><mi>i</mi> <mn>2</mn></msub><mi>⋯</mi><msub><mi>i</mi> <mi>n</mi></msub></mrow></msup><mspace width="thinmathspace"></mspace><msub><mi>a</mi> <mrow><msub><mi>i</mi> <mn>1</mn></msub><msub><mi>j</mi> <mn>1</mn></msub></mrow></msub><msub><mi>a</mi> <mrow><msub><mi>i</mi> <mn>2</mn></msub><msub><mi>j</mi> <mn>2</mn></msub></mrow></msub><mi>⋯</mi><msub><mi>a</mi> <mrow><msub><mi>i</mi> <mi>n</mi></msub><msub><mi>j</mi> <mi>n</mi></msub></mrow></msub><mspace width="thinmathspace"></mspace><msup><mi>ϵ</mi> <mrow><msub><mi>j</mi> <mn>1</mn></msub><msub><mi>j</mi> <mn>2</mn></msub><mi>⋯</mi><msub><mi>j</mi> <mi>n</mi></msub></mrow></msup></mrow><annotation encoding="application/x-tex"> \det(A) \;=\; \frac{1}{n!} \epsilon^{ i_1 i_2 \cdots i_n } \, a_{i_1 j_1} a_{i_2 j_2} \cdots a_{i_n j_n} \, \epsilon^{j_1 j_2 \cdots j_n} </annotation></semantics></math></div> <h2 id="properties">Properties</h2> <p>We work over <a class="existingWikiWord" href="/nlab/show/fields">fields</a> of arbitrary <a class="existingWikiWord" href="/nlab/show/characteristic">characteristic</a>. The determinant satisfies the following properties, which taken together uniquely characterize the determinant. Write a square <a class="existingWikiWord" href="/nlab/show/matrix">matrix</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> as a row of column <a class="existingWikiWord" href="/nlab/show/vectors">vectors</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>v</mi> <mn>1</mn></msub><mo>,</mo><mi>…</mi><mo>,</mo><msub><mi>v</mi> <mi>n</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(v_1, \ldots, v_n)</annotation></semantics></math>.</p> <ol> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>det</mi></mrow><annotation encoding="application/x-tex">\det</annotation></semantics></math> is separately linear in each column vector:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>det</mi><mo stretchy="false">(</mo><msub><mi>v</mi> <mn>1</mn></msub><mo>,</mo><mi>…</mi><mo>,</mo><mi>a</mi><mi>v</mi><mo>+</mo><mi>b</mi><mi>w</mi><mo>,</mo><mi>…</mi><mo>,</mo><msub><mi>v</mi> <mi>n</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mi>a</mi><mi>det</mi><mo stretchy="false">(</mo><msub><mi>v</mi> <mn>1</mn></msub><mo>,</mo><mi>…</mi><mo>,</mo><mi>v</mi><mo>,</mo><mi>…</mi><mo>,</mo><msub><mi>v</mi> <mi>n</mi></msub><mo stretchy="false">)</mo><mo>+</mo><mi>b</mi><mi>det</mi><mo stretchy="false">(</mo><msub><mi>v</mi> <mn>1</mn></msub><mo>,</mo><mi>…</mi><mo>,</mo><mi>w</mi><mo>,</mo><mi>…</mi><mo>,</mo><msub><mi>v</mi> <mi>n</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\det(v_1, \ldots, a v + b w, \ldots, v_n) = a\det(v_1, \ldots, v, \ldots, v_n) + b\det(v_1, \ldots, w, \ldots, v_n)</annotation></semantics></math></div></li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>det</mi><mo stretchy="false">(</mo><msub><mi>v</mi> <mn>1</mn></msub><mo>,</mo><mi>…</mi><mo>,</mo><msub><mi>v</mi> <mi>n</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\det(v_1, \ldots, v_n) = 0</annotation></semantics></math> whenever <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>v</mi> <mi>i</mi></msub><mo>=</mo><msub><mi>v</mi> <mi>j</mi></msub></mrow><annotation encoding="application/x-tex">v_i = v_j</annotation></semantics></math> for distinct <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo>,</mo><mi>j</mi></mrow><annotation encoding="application/x-tex">i, j</annotation></semantics></math>.</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>det</mi><mo stretchy="false">(</mo><mi>I</mi><mo stretchy="false">)</mo><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">\det(I) = 1</annotation></semantics></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi></mrow><annotation encoding="application/x-tex">I</annotation></semantics></math> is the identity matrix.</p> </li> </ol> <p>Other properties may be worked out, starting from the explicit formula or otherwise:</p> <ul> <li> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> is a diagonal matrix, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>det</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\det(A)</annotation></semantics></math> is the product of its diagonal entries.</p> </li> <li> <p>More generally, if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> is an upper (or lower) triangular matrix, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>det</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\det(A)</annotation></semantics></math> is the product of the diagonal entries.</p> </li> <li> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><mo stretchy="false">/</mo><mi>k</mi></mrow><annotation encoding="application/x-tex">E/k</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/field+extension">field extension</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> is a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>-linear map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi><mo>→</mo><mi>V</mi></mrow><annotation encoding="application/x-tex">V \to V</annotation></semantics></math>, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>det</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo>=</mo><mi>det</mi><mo stretchy="false">(</mo><mi>E</mi><msub><mo>⊗</mo> <mi>k</mi></msub><mi>f</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\det(f) = \det(E \otimes_k f)</annotation></semantics></math>. Using the preceding properties and the <span class="newWikiWord">Jordan normal form<a href="/nlab/new/Jordan+normal+form">?</a></span> of a matrix, this means that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>det</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\det(f)</annotation></semantics></math> is the product of its <a class="existingWikiWord" href="/nlab/show/eigenvalues">eigenvalues</a> (counted with multiplicity), as computed in the <a class="existingWikiWord" href="/nlab/show/algebraic+closure">algebraic closure</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>.</p> </li> <li> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>A</mi> <mi>t</mi></msup></mrow><annotation encoding="application/x-tex">A^t</annotation></semantics></math> is the transpose of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>det</mi><mo stretchy="false">(</mo><msup><mi>A</mi> <mi>t</mi></msup><mo stretchy="false">)</mo><mo>=</mo><mi>det</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\det(A^t) = \det(A)</annotation></semantics></math>.</p> </li> </ul> <h3 id="cramers_rule">Cramer’s rule</h3> <p>A simple observation which flows from these basic properties is</p> <div class="num_prop"> <h6 id="proposition_2">Proposition</h6> <p><strong>(Cramer’s Rule)</strong></p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>v</mi> <mn>1</mn></msub><mo>,</mo><mi>…</mi><mo>,</mo><msub><mi>v</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">v_1, \ldots, v_n</annotation></semantics></math> be column vectors of dimension <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>, and suppose</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>w</mi><mo>=</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mi>j</mi></munder><msub><mi>a</mi> <mi>j</mi></msub><msub><mi>v</mi> <mi>j</mi></msub><mo>.</mo></mrow><annotation encoding="application/x-tex">w = \sum_j a_j v_j.</annotation></semantics></math></div> <p>Then for each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math> we have</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>a</mi> <mi>j</mi></msub><mi>det</mi><mo stretchy="false">(</mo><msub><mi>v</mi> <mn>1</mn></msub><mo>,</mo><mi>…</mi><mo>,</mo><msub><mi>v</mi> <mi>i</mi></msub><mo>,</mo><mi>…</mi><mo>,</mo><msub><mi>v</mi> <mi>n</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mi>det</mi><mo stretchy="false">(</mo><msub><mi>v</mi> <mn>1</mn></msub><mo>,</mo><mi>…</mi><mo>,</mo><mi>w</mi><mo>,</mo><mi>…</mi><mo>,</mo><msub><mi>v</mi> <mi>n</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">a_j \det(v_1, \ldots, v_i, \ldots, v_n) = \det(v_1, \ldots, w, \ldots, v_n)</annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>w</mi></mrow><annotation encoding="application/x-tex">w</annotation></semantics></math> occurs as the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>i</mi> <mi>th</mi></msup></mrow><annotation encoding="application/x-tex">i^{th}</annotation></semantics></math> column vector on the right.</p> </div> <div class="proof"> <h6 id="proof_3">Proof</h6> <p>This follows straightforwardly from properties 1 and 2 above.</p> </div> <p>For instance, given a square matrix <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>det</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>≠</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\det(A) \neq 0</annotation></semantics></math>, and writing <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>=</mo><mo stretchy="false">(</mo><msub><mi>v</mi> <mn>1</mn></msub><mo>,</mo><mi>…</mi><mo>,</mo><msub><mi>v</mi> <mi>n</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">A = (v_1, \ldots, v_n)</annotation></semantics></math>, this allows us to solve for a vector <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi></mrow><annotation encoding="application/x-tex">a</annotation></semantics></math> in an equation</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>⋅</mo><mi>a</mi><mo>=</mo><mi>w</mi></mrow><annotation encoding="application/x-tex">A \cdot a = w</annotation></semantics></math></div> <p>and we easily conclude that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> is invertible if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>det</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>≠</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\det(A) \neq 0</annotation></semantics></math>.</p> <div class="num_remark"> <h6 id="remark_2">Remark</h6> <p>This holds true even if we replace the field <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math> by an arbitrary commutative <a class="existingWikiWord" href="/nlab/show/ring">ring</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>, and we replace the condition <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>det</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>≠</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\det(A) \neq 0</annotation></semantics></math> by the condition that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>det</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\det(A)</annotation></semantics></math> is a unit. (The entire development given above goes through, <em>mutatis mutandis</em>.)</p> </div> <h3 id="characteristic_polynomial_and_cayleyhamilton_theorem">Characteristic polynomial and Cayley-Hamilton theorem</h3> <p>Given a linear endomorphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>M</mi><mo>→</mo><mi>M</mi></mrow><annotation encoding="application/x-tex">f: M\to M</annotation></semantics></math> of a finite rank free unital module over a commutative unital ring, one can consider the zeros of the <a class="existingWikiWord" href="/nlab/show/characteristic+polynomial">characteristic polynomial</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>det</mi><mo stretchy="false">(</mo><mi>t</mi><mo>⋅</mo><msub><mn>1</mn> <mi>V</mi></msub><mo>−</mo><mi>f</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\det(t \cdot 1_V - f)</annotation></semantics></math>. The coefficients of the polynomial are the concern of the <a class="existingWikiWord" href="/nlab/show/Cayley-Hamilton+theorem">Cayley-Hamilton theorem</a>.</p> <h3 id="over_the_real_numbers_volume_and_orientation">Over the real numbers: volume and orientation</h3> <p>A useful intuition to have for determinants of <a class="existingWikiWord" href="/nlab/show/real+number">real</a> matrices is that they measure <em>change of volume</em>. That is, an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>×</mo><mi>n</mi></mrow><annotation encoding="application/x-tex">n \times n</annotation></semantics></math> matrix with real entries will map a standard unit cube in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^n</annotation></semantics></math> to a parallelpiped in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^n</annotation></semantics></math> (quashed to lie in a hyperplane if the matrix is singular), and the determinant is, up to sign, the volume of the parallelpiped. It is easy to convince oneself of this in the planar case by a simple dissection of a parallelogram, rearranging the dissected pieces in the style of Euclid to form a rectangle. In algebraic terms, the dissection and rearrangement amount to applying shearing or <a class="existingWikiWord" href="/nlab/show/elementary+column+operations">elementary column operations</a> to the matrix which, by the properties discussed earlier, leave the determinant unchanged. These operations transform the matrix into a diagonal matrix whose determinant is the area of the corresponding rectangle. This procedure easily generalizes to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math> dimensions.</p> <p>The sign itself is a matter of interest. An invertible transformation <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><mi>V</mi><mo>→</mo><mi>V</mi></mrow><annotation encoding="application/x-tex">f \colon V \to V</annotation></semantics></math> is said to be <strong><a class="existingWikiWord" href="/nlab/show/orientation">orientation</a>-preserving</strong> if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>det</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\det(f)</annotation></semantics></math> is positive, and <strong>orientation-reversing</strong> if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>det</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\det(f)</annotation></semantics></math> is negative. Orientations play an important role throughout geometry and algebraic topology, for example in the study of orientable manifolds (where the tangent bundle as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>GL</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">GL(n)</annotation></semantics></math>-bundle can be lifted to a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>GL</mi> <mo>+</mo></msub><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">GL_+(n)</annotation></semantics></math>-bundle structure, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>GL</mi> <mo>+</mo></msub><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>↪</mo><mi>GL</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">GL_+(n) \hookrightarrow GL(n)</annotation></semantics></math> being the subgroup of matrices of positive determinant). See also <a class="existingWikiWord" href="/nlab/show/KO-theory">KO-theory</a>.</p> <p>Finally, we include one more property of determinants which pertains to matrices with real coefficients (which works slightly more generally for matrices with coefficients in a <a class="existingWikiWord" href="/nlab/show/local+field">local field</a>):</p> <h3 id="AsAPolynomialInTracesofPowers">As a polynomial in traces of powers</h3> <p>On the <a class="existingWikiWord" href="/nlab/show/relation+between+determinant+and+trace">relation between determinant and trace</a>:</p> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> is an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>×</mo><mi>n</mi></mrow><annotation encoding="application/x-tex">n \times n</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/matrix">matrix</a>, the determinant of its <a class="existingWikiWord" href="/nlab/show/exponential">exponential</a> equals the <a class="existingWikiWord" href="/nlab/show/exponential">exponential</a> of its <a class="existingWikiWord" href="/nlab/show/trace">trace</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>det</mi><mo stretchy="false">(</mo><mi>exp</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><mi>exp</mi><mo stretchy="false">(</mo><mi>tr</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \det(\exp(A)) = \exp(tr(A)) \,. </annotation></semantics></math></div> <p>More generally, the determinant of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/polynomial">polynomial</a> in the <a class="existingWikiWord" href="/nlab/show/traces">traces</a> of the <a class="existingWikiWord" href="/nlab/show/powers">powers</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>:</p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>2</mn><mo>×</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">2 \times 2</annotation></semantics></math>-matrices:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>det</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mrow><mo>(</mo><mi>tr</mi><mo stretchy="false">(</mo><mi>A</mi><msup><mo stretchy="false">)</mo> <mn>2</mn></msup><mo>−</mo><mi>tr</mi><mo stretchy="false">(</mo><msup><mi>A</mi> <mn>2</mn></msup><mo stretchy="false">)</mo><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex"> det(A) \;=\; \tfrac{1}{2}\left( tr(A)^2 - tr(A^2) \right) </annotation></semantics></math></div> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>3</mn><mo>×</mo><mn>3</mn></mrow><annotation encoding="application/x-tex">3 \times 3</annotation></semantics></math>-matrices:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>det</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>6</mn></mfrac></mstyle><mrow><mo>(</mo><mo stretchy="false">(</mo><mi>tr</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><msup><mo stretchy="false">)</mo> <mn>3</mn></msup><mo>−</mo><mn>3</mn><mi>tr</mi><mo stretchy="false">(</mo><msup><mi>A</mi> <mn>2</mn></msup><mo stretchy="false">)</mo><mi>tr</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>+</mo><mi>tr</mi><mo stretchy="false">(</mo><msup><mi>A</mi> <mn>3</mn></msup><mo stretchy="false">)</mo><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex"> det(A) \;=\; \tfrac{1}{6} \left( (tr(A))^3 - 3 tr(A^2) tr(A) + tr(A^3) \right) </annotation></semantics></math></div> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>4</mn><mo>×</mo><mn>4</mn></mrow><annotation encoding="application/x-tex">4 \times 4</annotation></semantics></math>-matrices:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>det</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>24</mn></mfrac></mstyle><mrow><mo>(</mo><mo stretchy="false">(</mo><mi>tr</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><msup><mo stretchy="false">)</mo> <mn>4</mn></msup><mo>−</mo><mn>6</mn><mi>tr</mi><mo stretchy="false">(</mo><msup><mi>A</mi> <mn>2</mn></msup><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>tr</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><msup><mo stretchy="false">)</mo> <mn>2</mn></msup><mo>+</mo><mn>3</mn><mo stretchy="false">(</mo><mi>tr</mi><mo stretchy="false">(</mo><msup><mi>A</mi> <mn>2</mn></msup><mo stretchy="false">)</mo><msup><mo stretchy="false">)</mo> <mn>2</mn></msup><mo>+</mo><mn>8</mn><mi>tr</mi><mo stretchy="false">(</mo><msup><mi>A</mi> <mn>3</mn></msup><mo stretchy="false">)</mo><mi>tr</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>−</mo><mn>6</mn><mi>tr</mi><mo stretchy="false">(</mo><msup><mi>A</mi> <mn>4</mn></msup><mo stretchy="false">)</mo><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex"> det(A) \;=\; \tfrac{1}{24} \left( (tr(A))^4 - 6 tr(A^2)(tr(A))^2 + 3 (tr(A^2))^2 + 8 tr(A^3) tr(A) - 6 tr(A^4) \right) </annotation></semantics></math></div> <p>Generally for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>×</mo><mi>n</mi></mrow><annotation encoding="application/x-tex">n \times n</annotation></semantics></math>-matrices (<a href="#KondratyukKrivoruchenko92">Kondratyuk-Krivoruchenko 92, appendix B</a>):</p> <div class="maruku-equation" id="eq:DeterminantAsPolynomialInTracesOfPowers"><span class="maruku-eq-number">(3)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>det</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo><mfrac linethickness="0"><mrow><mrow><msub><mi>k</mi> <mn>1</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>k</mi> <mi>n</mi></msub><mo>∈</mo><mi>ℕ</mi></mrow></mrow><mrow><mrow><munderover><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo><mrow><mi>ℓ</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><mi>ℓ</mi><msub><mi>k</mi> <mi>ℓ</mi></msub><mo>=</mo><mi>n</mi></mrow></mrow></mfrac></munder><munderover><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo><mrow><mi>l</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><mfrac><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><msup><mo stretchy="false">)</mo> <mrow><msub><mi>k</mi> <mi>l</mi></msub><mo>+</mo><mn>1</mn></mrow></msup></mrow><mrow><msup><mi>l</mi> <mrow><msub><mi>k</mi> <mi>l</mi></msub></mrow></msup><msub><mi>k</mi> <mi>l</mi></msub><mo>!</mo></mrow></mfrac><msup><mrow><mo>(</mo><mi>tr</mi><mo stretchy="false">(</mo><msup><mi>A</mi> <mi>l</mi></msup><mo stretchy="false">)</mo><mo>)</mo></mrow> <mrow><msub><mi>k</mi> <mi>l</mi></msub></mrow></msup></mrow><annotation encoding="application/x-tex"> det(A) \;=\; \underset{ { k_1,\cdots, k_n \in \mathbb{N} } \atop { \underoverset{\ell = 1}{n}{\sum} \ell k_\ell = n } }{\sum} \underoverset{ l = 1 }{ n }{\prod} \frac{ (-1)^{k_l + 1} }{ l^{k_l} k_l ! } \left(tr(A^l)\right)^{k_l} </annotation></semantics></math></div> <div class="proof"> <h6 id="proof_of_">Proof of <a class="maruku-eqref" href="#eq:DeterminantAsPolynomialInTracesOfPowers">(3)</a></h6> <p>It is enough to prove this for semisimple matrices <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> (matrices that are <a class="existingWikiWord" href="/nlab/show/diagonalizable+matrix">diagonalizable</a> upon passing to the <a class="existingWikiWord" href="/nlab/show/algebraic+closure">algebraic closure</a> of the ground field) because this <a class="existingWikiWord" href="/nlab/show/subset">subset</a> of matrices is <a class="existingWikiWord" href="/nlab/show/Zariski+topology">Zariski</a> <a class="existingWikiWord" href="/nlab/show/dense+subset">dense</a> (using for example the nonvanishing of the <a class="existingWikiWord" href="/nlab/show/discriminant">discriminant</a> of the <a class="existingWikiWord" href="/nlab/show/characteristic+polynomial">characteristic polynomial</a>) and the set of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> for which the equation holds is <a class="existingWikiWord" href="/nlab/show/Zariski+topology">Zariski</a> <a class="existingWikiWord" href="/nlab/show/closed+subset">closed</a>.</p> <p>Thus, without loss of generality we may suppose that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/diagonal+matrix">diagonal</a> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/eigenvalues">eigenvalues</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>λ</mi> <mn>1</mn></msub><mo>,</mo><mi>…</mi><mo>,</mo><msub><mi>λ</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">\lambda_1, \ldots, \lambda_n</annotation></semantics></math> along the diagonal, where the statement can be rewritten as follows. Letting <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>p</mi> <mi>k</mi></msub><mo>=</mo><mi>tr</mi><mo stretchy="false">(</mo><msup><mi>A</mi> <mi>k</mi></msup><mo stretchy="false">)</mo><mo>=</mo><msubsup><mi>λ</mi> <mn>1</mn> <mi>k</mi></msubsup><mo>+</mo><mi>…</mi><mo>+</mo><msubsup><mi>λ</mi> <mi>n</mi> <mi>k</mi></msubsup></mrow><annotation encoding="application/x-tex">p_k = tr(A^k) = \lambda_1^k + \ldots + \lambda_n^k</annotation></semantics></math>, the following identity holds:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><munderover><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow> <mi>n</mi></munderover><msub><mi>λ</mi> <mi>i</mi></msub><mo>=</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo><mfrac linethickness="0"><mrow><mrow><msub><mi>k</mi> <mn>1</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>k</mi> <mi>n</mi></msub><mo>∈</mo><mi>ℕ</mi></mrow></mrow><mrow><mrow><munderover><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo><mrow><mi>ℓ</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><mi>ℓ</mi><msub><mi>k</mi> <mi>ℓ</mi></msub><mo>=</mo><mi>n</mi></mrow></mrow></mfrac></munder><munderover><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo><mrow><mi>l</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><mfrac><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><msup><mo stretchy="false">)</mo> <mrow><msub><mi>k</mi> <mi>l</mi></msub><mo>+</mo><mn>1</mn></mrow></msup></mrow><mrow><msup><mi>l</mi> <mrow><msub><mi>k</mi> <mi>l</mi></msub></mrow></msup><msub><mi>k</mi> <mi>l</mi></msub><mo>!</mo></mrow></mfrac><msubsup><mi>p</mi> <mi>l</mi> <mrow><msub><mi>k</mi> <mi>l</mi></msub></mrow></msubsup></mrow><annotation encoding="application/x-tex"> \prod_{i=1}^n \lambda_i = \underset{ { k_1,\cdots, k_n \in \mathbb{N} } \atop { \underoverset{\ell = 1}{n}{\sum} \ell k_\ell = n } }{\sum} \underoverset{ l = 1 }{ n }{\prod} \frac{ (-1)^{k_l + 1} }{ l^{k_l} k_l ! } p_l^{k_l} </annotation></semantics></math></div> <p>This of course is just a <a class="existingWikiWord" href="/nlab/show/polynomial">polynomial</a> identity, one closely related to various of the <a class="existingWikiWord" href="/nlab/show/Newton+identities">Newton identities</a> that concern symmetric polynomials in indeterminates <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>x</mi> <mn>1</mn></msub><mo>,</mo><mi>…</mi><mo>,</mo><msub><mi>x</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">x_1, \ldots, x_n</annotation></semantics></math>. Thus we again let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>p</mi> <mi>k</mi></msub><mo>=</mo><msubsup><mi>x</mi> <mn>1</mn> <mi>k</mi></msubsup><mo>+</mo><mi>…</mi><mo>+</mo><msubsup><mi>x</mi> <mi>n</mi> <mi>k</mi></msubsup></mrow><annotation encoding="application/x-tex">p_k = x_1^k + \ldots + x_n^k</annotation></semantics></math>, and define the <a class="existingWikiWord" href="/nlab/show/elementary+symmetric+polynomials">elementary symmetric polynomials</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>σ</mi> <mi>k</mi></msub><mo>=</mo><msub><mi>σ</mi> <mi>k</mi></msub><mo stretchy="false">(</mo><msub><mi>x</mi> <mn>1</mn></msub><mo>,</mo><mi>…</mi><mo>,</mo><msub><mi>x</mi> <mi>n</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\sigma_k = \sigma_k(x_1, \ldots, x_n)</annotation></semantics></math> via the generating function identity</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>k</mi><mo>≥</mo><mn>0</mn></mrow></munder><msub><mi>σ</mi> <mi>k</mi></msub><msup><mi>t</mi> <mi>k</mi></msup><mo>=</mo><munderover><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow> <mi>n</mi></munderover><mo stretchy="false">(</mo><mn>1</mn><mo>+</mo><msub><mi>x</mi> <mi>i</mi></msub><mi>t</mi><mo stretchy="false">)</mo><mo>.</mo></mrow><annotation encoding="application/x-tex"> \sum_{k \geq 0} \sigma_k t^k = \prod_{i=1}^n (1 + x_i t). </annotation></semantics></math></div> <p>Then we compute</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>k</mi><mo>≥</mo><mn>0</mn></mrow></munder><msub><mi>σ</mi> <mi>k</mi></msub><msup><mi>t</mi> <mi>k</mi></msup></mtd> <mtd><mo>=</mo></mtd> <mtd><munderover><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow> <mi>n</mi></munderover><mo stretchy="false">(</mo><mn>1</mn><mo>+</mo><msub><mi>x</mi> <mi>i</mi></msub><mi>t</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo></mtd> <mtd><mi>exp</mi><mrow><mo>(</mo><munderover><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow> <mi>n</mi></munderover><mi>log</mi><mo stretchy="false">(</mo><mn>1</mn><mo>+</mo><msub><mi>x</mi> <mi>i</mi></msub><mi>t</mi><mo stretchy="false">)</mo><mo>)</mo></mrow></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo></mtd> <mtd><mi>exp</mi><mrow><mo>(</mo><munderover><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow> <mi>n</mi></munderover><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>k</mi><mo>≥</mo><mn>1</mn></mrow></munder><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><msup><mo stretchy="false">)</mo> <mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msup><mfrac><mrow><msubsup><mi>x</mi> <mi>i</mi> <mi>k</mi></msubsup></mrow><mi>k</mi></mfrac><msup><mi>t</mi> <mi>k</mi></msup><mo>)</mo></mrow></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo></mtd> <mtd><mi>exp</mi><mrow><mo>(</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>k</mi><mo>≥</mo><mn>1</mn></mrow></munder><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><msup><mo stretchy="false">)</mo> <mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msup><mfrac><mrow><msub><mi>p</mi> <mi>k</mi></msub></mrow><mi>k</mi></mfrac><msup><mi>t</mi> <mi>k</mi></msup><mo>)</mo></mrow></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex">\array{ \sum_{k \geq 0} \sigma_k t^k & = & \prod_{i=1}^n (1 + x_i t) \\ & = & \exp\left(\sum_{i=1}^n \log(1 + x_i t)\right) \\ & = & \exp\left(\sum_{i=1}^n \sum_{k \geq 1} (-1)^{k+1} \frac{x_i^k}{k} t^k \right)\\ & = & \exp\left( \sum_{k \geq 1} (-1)^{k+1} \frac{p_k}{k} t^k\right) } </annotation></semantics></math></div> <p>and simply match coefficients of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>t</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">t^n</annotation></semantics></math> in the initial and final series expansions, where we easily compute</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>x</mi> <mn>1</mn></msub><msub><mi>x</mi> <mn>2</mn></msub><mi>…</mi><msub><mi>x</mi> <mi>n</mi></msub><mo>=</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>n</mi><mo>=</mo><msub><mi>k</mi> <mn>1</mn></msub><mo>+</mo><mn>2</mn><msub><mi>k</mi> <mn>2</mn></msub><mo>+</mo><mi>…</mi><mo>+</mo><mi>n</mi><msub><mi>k</mi> <mi>n</mi></msub></mrow></munder><munderover><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>l</mi><mo>=</mo><mn>1</mn></mrow> <mi>n</mi></munderover><mfrac><mn>1</mn><mrow><mo stretchy="false">(</mo><msub><mi>k</mi> <mi>l</mi></msub><mo stretchy="false">)</mo><mo>!</mo></mrow></mfrac><msup><mrow><mo>(</mo><mfrac><mrow><msub><mi>p</mi> <mi>l</mi></msub></mrow><mi>l</mi></mfrac><mo>)</mo></mrow> <mrow><msub><mi>k</mi> <mi>l</mi></msub></mrow></msup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><msup><mo stretchy="false">)</mo> <mrow><msub><mi>k</mi> <mi>l</mi></msub><mo>+</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex"> x_1 x_2 \ldots x_n = \sum_{n = k_1 + 2k_2 + \ldots + n k_n} \prod_{l=1}^n \frac1{(k_l)!} \left(\frac{p_l}{l}\right)^{k_l} (-1)^{k_l+1} </annotation></semantics></math></div> <p>This completes the proof.</p> </div> <h2 id="in_terms_of_berezinian_integrals">In terms of Berezinian integrals</h2> <p>see <a class="existingWikiWord" href="/nlab/show/Pfaffian">Pfaffian</a> for the moment</p> <h2 id="related_entries">Related entries</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/matrix">matrix</a>, <a class="existingWikiWord" href="/nlab/show/linear+algebra">linear algebra</a>, <a class="existingWikiWord" href="/nlab/show/exterior+algebra">exterior algebra</a>, <a class="existingWikiWord" href="/nlab/show/characteristic+polynomial">characteristic polynomial</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/principal+submatrix">principal submatrix</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quasideterminant">quasideterminant</a>, <a class="existingWikiWord" href="/nlab/show/Berezinian">Berezinian</a>,<a class="existingWikiWord" href="/nlab/show/Jacobian">Jacobian</a>, <a class="existingWikiWord" href="/nlab/show/Pfaffian">Pfaffian</a>, <a class="existingWikiWord" href="/nlab/show/hafnian">hafnian</a>, <a class="existingWikiWord" href="/nlab/show/Wronskian">Wronskian</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Vandermonde+determinant">Vandermonde determinant</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Dieudonn%C3%A9+determinant">Dieudonné determinant</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/resultant">resultant</a>, <a class="existingWikiWord" href="/nlab/show/discriminant">discriminant</a>, <a class="existingWikiWord" href="/nlab/show/hyperdeterminant">hyperdeterminant</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/functional+determinant">functional determinant</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/determinant+line">determinant line</a>, <a class="existingWikiWord" href="/nlab/show/determinant+line+bundle">determinant line bundle</a>, <a class="existingWikiWord" href="/nlab/show/Pfaffian+line+bundle">Pfaffian line bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/density+bundle">density bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pseudoscalar">pseudoscalar</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Slater+determinant">Slater determinant</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/permanent">permanent</a></p> </li> </ul> <h2 id="references">References</h2> <p>See also</p> <ul> <li>Wikipedia, <em><a href="https://en.wikipedia.org/wiki/Determinant">Determinant</a></em></li> </ul> <p>One derivation of the formula <a class="maruku-eqref" href="#eq:DeterminantAsPolynomialInTracesOfPowers">(3)</a> for the determinant as a polynomial in traces of powers is spelled out in appendix B of</p> <ul> <li id="KondratyukKrivoruchenko92">L. A. Kondratyuk, I. Krivoruchenko, <em>Superconducting quark matter in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SU</mi><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">SU(2)</annotation></semantics></math> colour group</em>, Zeitschrift für Physik A Hadrons and Nuclei March 1992, Volume 344, Issue 1, pp 99–115 (<a href="https://doi.org/10.1007/BF01291027">doi:10.1007/BF01291027</a>)</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on January 8, 2025 at 07:59:53. 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