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Matrika enic - Wikipedija, prosta enciklopedija

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class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">skrij</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">Uvod</div> </a> </li> <li id="toc-Primeri" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Primeri"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Primeri</span> </div> </a> <ul id="toc-Primeri-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Lastnosti" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Lastnosti"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Lastnosti</span> </div> </a> <ul id="toc-Lastnosti-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Glej_tudi" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Glej_tudi"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Glej tudi</span> </div> </a> <ul id="toc-Glej_tudi-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Vsebina" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Vklopi kazalo vsebine" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Vklopi kazalo vsebine</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Matrika enic</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="P9jdi na članek v drugem jeziku. Na voljo v 17 jezikih." > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-17" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">17 jezikov</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Jedni%C4%8Dkov%C3%A1_matice" title="Jedničková matice – češčina" lang="cs" hreflang="cs" data-title="Jedničková matice" data-language-autonym="Čeština" data-language-local-name="češčina" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%9F%C4%95%D1%80%D1%80%D0%B5%D1%81%D0%B5%D0%BD_%D0%BC%D0%B0%D1%82%D1%80%D0%B8%D1%86%D0%B8" title="Пĕрресен матрици – čuvaščina" lang="cv" hreflang="cv" data-title="Пĕрресен матрици" data-language-autonym="Чӑвашла" data-language-local-name="čuvaščina" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Einsmatrix" title="Einsmatrix – nemščina" lang="de" hreflang="de" data-title="Einsmatrix" data-language-autonym="Deutsch" data-language-local-name="nemščina" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%A0%CE%AF%CE%BD%CE%B1%CE%BA%CE%B1%CF%82_%CE%B1%CF%80%CF%8C_%CE%BC%CE%BF%CE%BD%CE%AC%CE%B4%CE%B5%CF%82" title="Πίνακας από μονάδες – grščina" lang="el" hreflang="el" data-title="Πίνακας από μονάδες" data-language-autonym="Ελληνικά" data-language-local-name="grščina" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Matrix_of_ones" title="Matrix of ones – angleščina" lang="en" hreflang="en" data-title="Matrix of ones" data-language-autonym="English" data-language-local-name="angleščina" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%85%D8%A7%D8%AA%D8%B1%DB%8C%D8%B3_%DB%8C%DA%A9%E2%80%8C%D9%87%D8%A7" title="ماتریس یک‌ها – perzijščina" lang="fa" hreflang="fa" data-title="ماتریس یک‌ها" data-language-autonym="فارسی" data-language-local-name="perzijščina" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Ykk%C3%B6smatriisi" title="Ykkösmatriisi – finščina" lang="fi" hreflang="fi" data-title="Ykkösmatriisi" data-language-autonym="Suomi" data-language-local-name="finščina" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Matriz_de_uns" title="Matriz de uns – galicijščina" lang="gl" hreflang="gl" data-title="Matriz de uns" data-language-autonym="Galego" data-language-local-name="galicijščina" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%9E%D7%98%D7%A8%D7%99%D7%A6%D7%AA_%D7%90%D7%97%D7%93%D7%95%D7%AA" title="מטריצת אחדות – hebrejščina" lang="he" hreflang="he" data-title="מטריצת אחדות" data-language-autonym="עברית" data-language-local-name="hebrejščina" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Matrice_cu_toate_elementele_1" title="Matrice cu toate elementele 1 – romunščina" lang="ro" hreflang="ro" data-title="Matrice cu toate elementele 1" data-language-autonym="Română" data-language-local-name="romunščina" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9C%D0%B0%D1%82%D1%80%D0%B8%D1%86%D0%B0_%D0%B5%D0%B4%D0%B8%D0%BD%D0%B8%D1%86" title="Матрица единиц – ruščina" lang="ru" hreflang="ru" data-title="Матрица единиц" data-language-autonym="Русский" data-language-local-name="ruščina" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Matrica_e_nj%C3%ABsheve" title="Matrica e njësheve – albanščina" lang="sq" hreflang="sq" data-title="Matrica e njësheve" data-language-autonym="Shqip" data-language-local-name="albanščina" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%92%E0%AE%A9%E0%AF%8D%E0%AE%B1%E0%AF%81%E0%AE%95%E0%AE%B3%E0%AE%BF%E0%AE%A9%E0%AF%8D_%E0%AE%85%E0%AE%A3%E0%AE%BF" title="ஒன்றுகளின் அணி – tamilščina" lang="ta" hreflang="ta" data-title="ஒன்றுகளின் அணி" data-language-autonym="தமிழ்" data-language-local-name="tamilščina" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B9%80%E0%B8%A1%E0%B8%97%E0%B8%A3%E0%B8%B4%E0%B8%81%E0%B8%8B%E0%B9%8C%E0%B8%AB%E0%B8%99%E0%B8%B6%E0%B9%88%E0%B8%87" title="เมทริกซ์หนึ่ง – tajščina" lang="th" hreflang="th" data-title="เมทริกซ์หนึ่ง" data-language-autonym="ไทย" data-language-local-name="tajščina" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Birler_matrisi" title="Birler matrisi – turščina" lang="tr" hreflang="tr" data-title="Birler matrisi" data-language-autonym="Türkçe" data-language-local-name="turščina" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9C%D0%B0%D1%82%D1%80%D0%B8%D1%86%D1%8F_%D0%BE%D0%B4%D0%B8%D0%BD%D0%B8%D1%86%D1%8C" title="Матриця одиниць – ukrajinščina" lang="uk" hreflang="uk" data-title="Матриця одиниць" data-language-autonym="Українська" data-language-local-name="ukrajinščina" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E4%B8%80%E7%9F%A9%E9%99%A3" title="一矩陣 – kitajščina" lang="zh" hreflang="zh" data-title="一矩陣" data-language-autonym="中文" data-language-local-name="kitajščina" class="interlanguage-link-target"><span>中文</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit 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</div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">Iz Wikipedije, proste enciklopedije</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="sl" dir="ltr"><p><b>Matrika enic</b> je <a href="/wiki/Matrika" title="Matrika">matrika</a>, ki ima na vseh mestih enice (vrednost 1). Pri matrikah enic je pomembna razsežnost, ker matrike enic z različnimi razsežnostmi niso enake. Zaradi tega običajno ob oznaki zapišemo tudi razsežnost matrike. Matrika enic ni vedno kvadratna. </p> <div class="mw-heading mw-heading2"><h2 id="Primeri">Primeri</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Matrika_enic&amp;veaction=edit&amp;section=1" title="Uredi razdelek: Primeri" class="mw-editsection-visualeditor"><span>uredi</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Matrika_enic&amp;action=edit&amp;section=1" title="Urejanje izvorne kode razdelka: Primeri"><span>uredi kodo</span></a><span class="mw-editsection-bracket">]</span></span></div> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J_{2}={\begin{pmatrix}1&amp;1\\1&amp;1\end{pmatrix}};\quad J_{3}={\begin{pmatrix}1&amp;1&amp;1\\1&amp;1&amp;1\\1&amp;1&amp;1\end{pmatrix}};\quad J_{2,5}={\begin{pmatrix}1&amp;1&amp;1&amp;1&amp;1\\1&amp;1&amp;1&amp;1&amp;1\end{pmatrix}}.\quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>;</mo> <mspace width="1em" /> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>;</mo> <mspace width="1em" /> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>,</mo> <mn>5</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>.</mo> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle J_{2}={\begin{pmatrix}1&amp;1\\1&amp;1\end{pmatrix}};\quad J_{3}={\begin{pmatrix}1&amp;1&amp;1\\1&amp;1&amp;1\\1&amp;1&amp;1\end{pmatrix}};\quad J_{2,5}={\begin{pmatrix}1&amp;1&amp;1&amp;1&amp;1\\1&amp;1&amp;1&amp;1&amp;1\end{pmatrix}}.\quad }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b4bf223807ea94e2504b2ca94eb964c5f7627d9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:68.337ex; height:9.176ex;" alt="{\displaystyle J_{2}={\begin{pmatrix}1&amp;1\\1&amp;1\end{pmatrix}};\quad J_{3}={\begin{pmatrix}1&amp;1&amp;1\\1&amp;1&amp;1\\1&amp;1&amp;1\end{pmatrix}};\quad J_{2,5}={\begin{pmatrix}1&amp;1&amp;1&amp;1&amp;1\\1&amp;1&amp;1&amp;1&amp;1\end{pmatrix}}.\quad }"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Lastnosti">Lastnosti</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Matrika_enic&amp;veaction=edit&amp;section=2" title="Uredi razdelek: Lastnosti" class="mw-editsection-visualeditor"><span>uredi</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Matrika_enic&amp;action=edit&amp;section=2" title="Urejanje izvorne kode razdelka: Lastnosti"><span>uredi kodo</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Sled_matrike" title="Sled matrike">sled matrike enic</a> in <a href="/wiki/Determinanta" title="Determinanta">determinanta</a> sta enaki 1, če je razsežnost matrike <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1\times 1\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>&#x00D7;<!-- × --></mo> <mn>1</mn> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1\times 1\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81babe957d8bfeb2c53e471c5f71163004a40a8a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.552ex; height:2.176ex;" alt="{\displaystyle 1\times 1\,}"></span>, v vseh ostalih primerih pa sta enaka 0.</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J^{k}=n^{k-1}J,{\mbox{ za }}k=1,2,\ldots .\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo>=</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>&#x2212;<!-- − --></mo> <mn>1</mn> </mrow> </msup> <mi>J</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>&#xA0;za&#xA0;</mtext> </mstyle> </mrow> <mi>k</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mo>&#x2026;<!-- … --></mo> <mo>.</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle J^{k}=n^{k-1}J,{\mbox{ za }}k=1,2,\ldots .\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/88e90175b419239c334b3b1fccd0ff3a00ab7527" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:29.005ex; height:3.009ex;" alt="{\displaystyle J^{k}=n^{k-1}J,{\mbox{ za }}k=1,2,\ldots .\,}"></span></li> <li><a href="/wiki/Rang_(linearna_algebra)" title="Rang (linearna algebra)">rang matrike</a> je enak 1</li> <li><a href="/wiki/Lastna_vrednost" title="Lastna vrednost">latne vrednosti</a> so enake enkrat 1 in n-1-krat pa 0.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Glej_tudi">Glej tudi</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Matrika_enic&amp;veaction=edit&amp;section=3" title="Uredi razdelek: Glej tudi" class="mw-editsection-visualeditor"><span>uredi</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Matrika_enic&amp;action=edit&amp;section=3" title="Urejanje izvorne kode razdelka: Glej tudi"><span>uredi kodo</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Seznam_vrst_matrik" title="Seznam vrst matrik">seznam vrst matrik</a></li></ul> <!-- NewPP limit report Parsed by mw‐api‐int.codfw.main‐55fb8d7f4f‐fzgzr Cached time: 20241110134815 Cache expiry: 2592000 Reduced expiry: false Complications: [] CPU time usage: 0.011 seconds Real time usage: 0.065 seconds Preprocessor visited node count: 29/1000000 Post‐expand include size: 0/2097152 bytes Template argument size: 0/2097152 bytes Highest expansion depth: 2/100 Expensive parser function count: 0/500 Unstrip recursion depth: 0/20 Unstrip post‐expand size: 108/5000000 bytes Number of Wikibase entities loaded: 0/400 --> <!-- Transclusion expansion time report (%,ms,calls,template) 100.00% 0.000 1 -total --> <!-- Saved in parser cache with key slwiki:pcache:idhash:265056-0!canonical and timestamp 20241110134815 and revision id 3923332. 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