English+GermanWikipedia - /page/entry.mu

<!DOCTYPE html>

<title>Diagonalisierbare Matrix</title>

</head>

<h1 id="firstHeading" class="firstHeading mw-first-heading">Diagonalisierbare Matrix</h1>

</header>

</div>

<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="de" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="de" dir="ltr">Als diagonalisierbare Matrix bezeichnet man im <a href="Teilgebiete_der_Mathematik" title="Teilgebiete der Mathematik">mathematischen Teilgebiet</a> der <a href="Lineare_Algebra" title="Lineare Algebra">linearen Algebra</a> eine quadratische <a href="Matrix_(Mathematik)" title="Matrix (Mathematik)">Matrix</a>, die <a href="%C3%84hnlichkeit_(Matrix)" title="Ähnlichkeit (Matrix)">ähnlich</a> zu einer <a href="Diagonalmatrix" title="Diagonalmatrix">Diagonalmatrix</a> ist. Sie lässt sich mittels eines <a href="Basiswechsel_(Vektorraum)" title="Basiswechsel (Vektorraum)">Basiswechsels</a> (also der <a href="Konjugation_(Gruppentheorie)" title="Konjugation (Gruppentheorie)">Konjugation</a> mit einer <a href="Regul%C3%A4re_Matrix" title="Reguläre Matrix">regulären Matrix</a>) in eine Diagonalmatrix transformieren.<a href="#cite_note-1">[1]</a> Das Konzept lässt sich auf <a href="Endomorphismus" title="Endomorphismus">Endomorphismen</a> übertragen.

<div class="mw-heading mw-heading2"><h2 id="Definition"> Definition</h2></div>

Eine quadratische <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\!\times \!n\!\ }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n\!\times \!n\!\ }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/585d6ecdad7036d27c85f1a0bbebf174039e2b0e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.049ex; height:1.676ex;" alt="{\displaystyle n\!\times \!n\!\ }" loading="lazy">-Matrix <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"> über einem <a href="K%C3%B6rper_(Algebra)" title="Körper (Algebra)">Körper</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"> heißt diagonalisierbar oder diagonalähnlich, wenn es eine <a href="Diagonalmatrix" title="Diagonalmatrix">Diagonalmatrix</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D_{A}\in K^{n\times n}}">

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle D_{A}\in K^{n\times n}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/61d8b53e9850f6ba76c4b69c63d72e667d52a7b5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.807ex; height:2.676ex;" alt="{\displaystyle D_{A}\in K^{n\times n}}" loading="lazy"> gibt, zu der sie <a href="%C3%84hnlichkeit_(Matrix)" title="Ähnlichkeit (Matrix)">ähnlich</a> ist, d. h., zu der eine <a href="Regul%C3%A4re_Matrix" title="Reguläre Matrix">reguläre Matrix</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S\in K^{n\times n}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S\in K^{n\times n}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/479de165df2ac36a8cf035e918adc4016d6c979a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.917ex; height:2.343ex;" alt="{\displaystyle S\in K^{n\times n}}" loading="lazy"> existiert mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D_{A}=S^{-1}AS.}">

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle D_{A}=S^{-1}AS.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/34873cbd2307b1d0b3b621e561f557f7eccaaaed.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.231ex; height:3.009ex;" alt="{\displaystyle D_{A}=S^{-1}AS.}" loading="lazy">

Ein <a href="Endomorphismus" title="Endomorphismus">Endomorphismus</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\colon \,V\to V}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f\colon \,V\to V}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/be88fcc0abf057bb30ab14afb815bb30c77ed410.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.888ex; height:2.509ex;" alt="{\displaystyle f\colon \,V\to V}" loading="lazy"> eines endlichdimensionalen <a href="Vektorraum" title="Vektorraum">Vektorraumes</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle V}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy"> heißt diagonalisierbar, falls eine <a href="Basis_(Vektorraum)" title="Basis (Vektorraum)">Basis</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle B}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/47136aad860d145f75f3eed3022df827cee94d7a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}" loading="lazy"> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle V}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy"> existiert, bezüglich der die <a href="Abbildungsmatrix" title="Abbildungsmatrix">Abbildungsmatrix</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{B}^{B}(f)}">

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M_{B}^{B}(f)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ecac2e8b3098f605a1c5651991c8eadf3015b17c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.066ex; height:3.176ex;" alt="{\displaystyle M_{B}^{B}(f)}" loading="lazy"> eine Diagonalmatrix ist. Die Basis <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle B}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/47136aad860d145f75f3eed3022df827cee94d7a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}" loading="lazy"> besteht dann aus <a href="Eigenvektor" class="mw-redirect" title="Eigenvektor">Eigenvektoren</a> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy">.

<div class="mw-heading mw-heading3"><h3 id="Orthogonal_bzw._unitär_diagonalisierbare_Matrix"> Orthogonal bzw. unitär diagonalisierbare Matrix</h3></div>

Eine Matrix <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\in \mathbb {R} ^{n\times n}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A\in \mathbb {R} ^{n\times n}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ebb63eca276c71633e0939f5c5d4ef4bfc565518.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.745ex; height:2.343ex;" alt="{\displaystyle A\in \mathbb {R} ^{n\times n}}" loading="lazy"> heißt orthogonal diagonalisierbar, wenn eine <a href="Orthogonale_Matrix" title="Orthogonale Matrix">orthogonale Matrix</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S\in \mathbb {R} ^{n\times n}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S\in \mathbb {R} ^{n\times n}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6adbe2deba338fe8b5da6b2fd48fd3e6cd24dc2b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.501ex; height:2.343ex;" alt="{\displaystyle S\in \mathbb {R} ^{n\times n}}" loading="lazy"> existiert, sodass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{-1}AS=S^{\mathsf {T}}AS}">

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S^{-1}AS=S^{\mathsf {T}}AS}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5f38d78d085ea23e6133c48bd634b1a6736582c2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:16.311ex; height:2.676ex;" alt="{\displaystyle S^{-1}AS=S^{\mathsf {T}}AS}" loading="lazy"> eine Diagonalmatrix ist, wobei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{\mathsf {T}}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S^{\mathsf {T}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/92d20bca4c2ff413c816b9e7c38cbf8a8ba31968.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.873ex; height:2.676ex;" alt="{\displaystyle S^{\mathsf {T}}}" loading="lazy"> die zu <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4611d85173cd3b508e67077d4a1252c9c05abca2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}" loading="lazy"> <a href="Transponierte_Matrix" title="Transponierte Matrix">transponierte Matrix</a> bezeichnet. Das trifft genau dann zu, wenn die Matrix <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"> <a href="Symmetrische_Matrix" title="Symmetrische Matrix">symmetrisch</a> ist.

Eine Matrix <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\in \mathbb {C} ^{n\times n}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A\in \mathbb {C} ^{n\times n}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/83b3023de9ab6f59e511d7c8ae72d03b64bcecbc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.745ex; height:2.343ex;" alt="{\displaystyle A\in \mathbb {C} ^{n\times n}}" loading="lazy"> heißt unitär diagonalisierbar, wenn eine <a href="Unit%C3%A4re_Matrix" title="Unitäre Matrix">unitäre Matrix</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S\in \mathbb {C} ^{n\times n}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S\in \mathbb {C} ^{n\times n}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e253e5c48624be8d033bb62916a5922647537ab9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.501ex; height:2.343ex;" alt="{\displaystyle S\in \mathbb {C} ^{n\times n}}" loading="lazy"> existiert, sodass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{-1}AS=S^{\mathsf {H}}AS}">

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S^{-1}AS=S^{\mathsf {H}}AS}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a26bc6a23e1a2705f26899ca4f25ebc1462dbbe6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:16.355ex; height:2.676ex;" alt="{\displaystyle S^{-1}AS=S^{\mathsf {H}}AS}" loading="lazy"> eine Diagonalmatrix ist, wobei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{\mathsf {H}}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S^{\mathsf {H}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0af8e35a7c69e8be92b62c372195bba6c0be20dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.917ex; height:2.676ex;" alt="{\displaystyle S^{\mathsf {H}}}" loading="lazy"> die zu <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4611d85173cd3b508e67077d4a1252c9c05abca2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}" loading="lazy"> <a href="Adjungierte_Matrix" title="Adjungierte Matrix">adjungierte Matrix</a> bezeichnet. Das trifft genau dann zu, wenn die Matrix <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"> <a href="Normale_Matrix" title="Normale Matrix">normal</a> ist.

In einem endlichdimensionalen <a href="Euklidischer_Vektorraum" class="mw-redirect" title="Euklidischer Vektorraum">euklidischen</a> bzw. <a href="Unit%C3%A4rer_Vektorraum" class="mw-redirect" title="Unitärer Vektorraum">unitären Vektorraum</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle V}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy"> heißt ein Endomorphismus <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\colon \,V\to V}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f\colon \,V\to V}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/be88fcc0abf057bb30ab14afb815bb30c77ed410.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.888ex; height:2.509ex;" alt="{\displaystyle f\colon \,V\to V}" loading="lazy"> orthogonal bzw. unitär diagonalisierbar, wenn eine <a href="Orthonormalbasis" title="Orthonormalbasis">Orthonormalbasis</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle B}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/47136aad860d145f75f3eed3022df827cee94d7a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}" loading="lazy"> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle V}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy"> existiert, sodass die Abbildungsmatrix <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{B}^{B}(f)}">

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M_{B}^{B}(f)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ecac2e8b3098f605a1c5651991c8eadf3015b17c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.066ex; height:3.176ex;" alt="{\displaystyle M_{B}^{B}(f)}" loading="lazy"> eine Diagonalmatrix ist. Dieser Fall liegt genau dann vor, wenn der Endomorphismus <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> <a href="Selbstadjungiert" class="mw-redirect" title="Selbstadjungiert">selbstadjungiert</a> bzw. <a href="Normaler_Operator" title="Normaler Operator">normal</a> ist.

<div class="mw-heading mw-heading2"><h2 id="Weitere_Charakterisierungen_der_Diagonalisierbarkeit">Weitere Charakterisierungen der Diagonalisierbarkeit</h2></div>

Sei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"> eine <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy">-dimensionale Matrix mit Einträgen aus einem Körper <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy">. Jede der folgenden sechs Bedingungen wird genau dann erfüllt, wenn <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"> diagonalisierbar ist.

<ol><li>Das <a href="Minimalpolynom_(Lineare_Algebra)" title="Minimalpolynom (Lineare Algebra)">Minimalpolynom</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu _{A}(\lambda )}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mu _{A}(\lambda )}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ab0f2609a5e103f596d86f966683ddb1391a0a7a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.031ex; height:2.843ex;" alt="{\displaystyle \mu _{A}(\lambda )}" loading="lazy"> zerfällt vollständig in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k\leq n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k\leq n}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/621f658bb51d7caac329d29e9bf435361813777f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.704ex; height:2.343ex;" alt="{\displaystyle k\leq n}" loading="lazy"> paarweise verschiedene Linearfaktoren: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu _{A}(\lambda )=\pm (\lambda -\lambda _{1})\cdot \dots \cdot (\lambda -\lambda _{k})}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mu _{A}(\lambda )=\pm (\lambda -\lambda _{1})\cdot \dots \cdot (\lambda -\lambda _{k})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7a77830864885dd68e3415ddae9510a9636c9981.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.882ex; height:2.843ex;" alt="{\displaystyle \mu _{A}(\lambda )=\pm (\lambda -\lambda _{1})\cdot \dots \cdot (\lambda -\lambda _{k})}" loading="lazy"> mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda _{i}\in K}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \lambda _{i}\in K}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5c64ce320b5fa51f921fbb7e7e873aed04d84e2c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.062ex; height:2.509ex;" alt="{\displaystyle \lambda _{i}\in K}" loading="lazy"></li>

<li>Das <a href="Charakteristisches_Polynom" title="Charakteristisches Polynom">charakteristische Polynom</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi _{A}(\lambda )}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \chi _{A}(\lambda )}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ee56eb70fc1b2f610b090caa6d491695276b0b0a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.084ex; height:2.843ex;" alt="{\displaystyle \chi _{A}(\lambda )}" loading="lazy"> zerfällt vollständig in Linearfaktoren und die <a href="Geometrische_Vielfachheit" class="mw-redirect" title="Geometrische Vielfachheit">geometrische Vielfachheit</a> entspricht der <a href="Algebraische_Vielfachheit" class="mw-redirect" title="Algebraische Vielfachheit">algebraischen Vielfachheit</a> für jeden Eigenwert <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda _{i}\in K}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \lambda _{i}\in K}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5c64ce320b5fa51f921fbb7e7e873aed04d84e2c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.062ex; height:2.509ex;" alt="{\displaystyle \lambda _{i}\in K}" loading="lazy">.</li>

<li>Es gibt eine Basis für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K^{n}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K^{n}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1d63366b3d00300e06eee81786182062b98775c5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.312ex; height:2.343ex;" alt="{\displaystyle K^{n}}" loading="lazy">, die aus Eigenvektoren für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"> besteht.</li>

<li>Die Summe der Dimensionen der jeweiligen <a href="Eigenraum" title="Eigenraum">Eigenräume</a> ist gleich <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy">: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{\lambda \in \sigma (A)}\dim(E_{\lambda }(A))=n}">

</mrow>

</munder>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sum _{\lambda \in \sigma (A)}\dim(E_{\lambda }(A))=n}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/383da4d24bdd67a358fb19c8fcb314a89eb6a2a7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:22.53ex; height:6.009ex;" alt="{\displaystyle \sum _{\lambda \in \sigma (A)}\dim(E_{\lambda }(A))=n}" loading="lazy">, wobei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma (A)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sigma (A)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0a507cac58b93e190d8f9ff7e470e6c28ff772d0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.882ex; height:2.843ex;" alt="{\displaystyle \sigma (A)}" loading="lazy"> das <a href="Eigenwertproblem#Spektrum_und_Vielfachheiten" class="mw-redirect" title="Eigenwertproblem">Spektrum</a> bezeichnet.</li>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K^{n}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1d63366b3d00300e06eee81786182062b98775c5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.312ex; height:2.343ex;" alt="{\displaystyle K^{n}}" loading="lazy"> ist die <a href="Direkte_Summe" title="Direkte Summe">direkte Summe</a> der jeweiligen Eigenräume: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K^{n}=\bigoplus _{\lambda \in \sigma (A)}E_{\lambda }(A)}">

<msup>

</mrow>

</msup>

</mrow>

</munder>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K^{n}=\bigoplus _{\lambda \in \sigma (A)}E_{\lambda }(A)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/669f56ba8dcc324c4a692cbf7080f5fa4b7588d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:18.763ex; height:6.009ex;" alt="{\displaystyle K^{n}=\bigoplus _{\lambda \in \sigma (A)}E_{\lambda }(A)}" loading="lazy">.</li></ol>

Ist zusätzlich <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"> <a href="Algebraischer_Abschluss" title="Algebraischer Abschluss">algebraisch abgeschlossen</a>, ist die Aussage äquivalent zu:

<ol><li value="6"> Alle Jordanblöcke der <a href="Jordansche_Normalform" title="Jordansche Normalform">Jordanschen Normalform</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J_{A}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle J_{A}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/daf9814034acbaa033f4db4ec3620cb5bb1f969a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.755ex; height:2.509ex;" alt="{\displaystyle J_{A}}" loading="lazy"> haben die Dimension 1.</li></ol>

<div class="mw-heading mw-heading2"><h2 id="Eigenschaften_einer_diagonalisierbaren_Matrix">Eigenschaften einer diagonalisierbaren Matrix</h2></div>

<ul><li>Die Diagonaleinträge von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D_{A}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle D_{A}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b78af4a9d2d8aaec85ea4e6b0f997a31def04311.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.389ex; height:2.509ex;" alt="{\displaystyle D_{A}}" loading="lazy"> zu einer diagonalisierbaren Matrix <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"> sind gerade die Eigenwerte von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy">.</li>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4611d85173cd3b508e67077d4a1252c9c05abca2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}" loading="lazy"> invertierbar ist, folgt, dass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (Se_{1},\ldots ,Se_{n})}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (Se_{1},\ldots ,Se_{n})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f17aa61ed8bf31e74d92a98839a173a4b8a7d525.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.426ex; height:2.843ex;" alt="{\displaystyle (Se_{1},\ldots ,Se_{n})}" loading="lazy"> <a href="Lineare_Unabh%C3%A4ngigkeit" title="Lineare Unabhängigkeit">linear unabhängig</a> ist.</li>

<li>Es ergibt sich die notwendige Bedingung, dass eine <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy">-dimensionale diagonalisierbare Matrix <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"> linear unabhängige Eigenvektoren haben muss. Der Raum, auf dem sie operiert, besitzt also eine Basis aus Eigenvektoren der Matrix. Diese Bedingung ist aber auch hinreichend, denn aus <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"> gefundenen linear unabhängigen Eigenvektoren von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"> mit den dazugehörigen Eigenwerten lassen sich geeignete <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D_{A}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle D_{A}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b78af4a9d2d8aaec85ea4e6b0f997a31def04311.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.389ex; height:2.509ex;" alt="{\displaystyle D_{A}}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4611d85173cd3b508e67077d4a1252c9c05abca2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}" loading="lazy"> ganz direkt konstruieren.</li></ul>

<dl><dd>Das Problem reduziert sich damit auf das Auffinden von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"> linear unabhängigen Eigenvektoren von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy">.</dd></dl>

<ul><li>Eine notwendige, aber nicht hinreichende Bedingung für Diagonalisierbarkeit ist, dass das <a href="Charakteristisches_Polynom" title="Charakteristisches Polynom">charakteristische Polynom</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi _{A}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \chi _{A}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/49baf1caaa804f2d77bfc7570d102ee4a3cafa26.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.92ex; height:2.009ex;" alt="{\displaystyle \chi _{A}}" loading="lazy"> vollständig in Linearfaktoren zerfällt: So ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A={\begin{pmatrix}0&1\\0&0\end{pmatrix}}}">

<mrow>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A={\begin{pmatrix}0&1\\0&0\end{pmatrix}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ff57095860713106e2e9aad71c1fd8409aeabebd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:13.662ex; height:6.176ex;" alt="{\displaystyle A={\begin{pmatrix}0&1\\0&0\end{pmatrix}}}" loading="lazy"> nicht diagonalisierbar, obwohl <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi _{A}(X)=X^{2}}">

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \chi _{A}(X)=X^{2}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/546d05099ddf7775304601f61c9e973d136e4720.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.859ex; height:3.176ex;" alt="{\displaystyle \chi _{A}(X)=X^{2}}" loading="lazy">. Eine hinreichende, aber nicht notwendige Bedingung für Diagonalisierbarkeit ist, dass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi _{A}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \chi _{A}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/49baf1caaa804f2d77bfc7570d102ee4a3cafa26.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.92ex; height:2.009ex;" alt="{\displaystyle \chi _{A}}" loading="lazy"> vollständig in paarweise verschiedene Linearfaktoren zerfällt: So ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A={\begin{pmatrix}1&0\\0&1\end{pmatrix}}}">

<mrow>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A={\begin{pmatrix}1&0\\0&1\end{pmatrix}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a3d70afbc7a5cc91c4e79b55e6880b63269ba926.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:13.662ex; height:6.176ex;" alt="{\displaystyle A={\begin{pmatrix}1&0\\0&1\end{pmatrix}}}" loading="lazy"> diagonalisierbar, obwohl <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi _{A}(X)=(X-1)^{2}}">

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \chi _{A}(X)=(X-1)^{2}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/622b944be6baaadb6f64c8921397570b4c4feebd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.654ex; height:3.176ex;" alt="{\displaystyle \chi _{A}(X)=(X-1)^{2}}" loading="lazy">.</li>

<li>Für eine diagonalisierbare Matrix ist die <a href="Geometrische_Vielfachheit" class="mw-redirect" title="Geometrische Vielfachheit">geometrische Vielfachheit</a> ihrer <a href="Eigenwert" class="mw-redirect" title="Eigenwert">Eigenwerte</a> gleich der jeweiligen <a href="Algebraische_Vielfachheit" class="mw-redirect" title="Algebraische Vielfachheit">algebraischen Vielfachheit</a>. Das bedeutet, die <a href="Dimension_(Mathematik)" title="Dimension (Mathematik)">Dimension</a> der einzelnen <a href="Eigenraum" title="Eigenraum">Eigenräume</a> stimmt jeweils mit der algebraischen Vielfachheit der entsprechenden Eigenwerte im <a href="Charakteristisches_Polynom" title="Charakteristisches Polynom">charakteristischen Polynom</a> der Matrix überein.</li>

<li>Die <a href="Matrixpotenz" title="Matrixpotenz">Matrixpotenz</a> einer diagonalisierbaren Matrix <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"> lässt sich berechnen durch</li></ul>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</msubsup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A^{n}=S\cdot D_{A}^{n}\cdot S^{-1}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/19b2e7879a9461ef587320f5eb7bf56303e3a12a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:18.161ex; height:3.343ex;" alt="{\displaystyle A^{n}=S\cdot D_{A}^{n}\cdot S^{-1}}" loading="lazy"></dd>

<dd>Die Potenz einer Diagonalmatrix erhält man durch Potenzieren der Diagonalelemente.</dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Diagonalisierung">Diagonalisierung</h2></div>

Ist eine Matrix <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"> diagonalisierbar, existiert eine Diagonalmatrix <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D_{A}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle D_{A}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b78af4a9d2d8aaec85ea4e6b0f997a31def04311.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.389ex; height:2.509ex;" alt="{\displaystyle D_{A}}" loading="lazy">, für die die Ähnlichkeitsbedingung erfüllt ist:

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle D_{A}=S^{-1}AS}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/29488cd6eaa1bbdb9755fca5435a6f7f4653df15.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.584ex; height:3.009ex;" alt="{\displaystyle D_{A}=S^{-1}AS}" loading="lazy"></dd></dl>

Zur Diagonalisierung dieser Matrix berechnet man die Diagonalmatrix <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D_{A}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle D_{A}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b78af4a9d2d8aaec85ea4e6b0f997a31def04311.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.389ex; height:2.509ex;" alt="{\displaystyle D_{A}}" loading="lazy"> und eine zugehörige Basis aus Eigenvektoren. Dies geschieht in drei Schritten:

<ol><li>Es werden die <a href="Eigenwertproblem" class="mw-redirect" title="Eigenwertproblem">Eigenwerte</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda _{i}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \lambda _{i}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/72fde940918edf84caf3d406cc7d31949166820f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.155ex; height:2.509ex;" alt="{\displaystyle \lambda _{i}}" loading="lazy"> der Matrix <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"> bestimmt. (Einzelne Eigenwerte können dabei mehrfach vorkommen.)</li>

<li>Es werden die <a href="Eigenraum" title="Eigenraum">Eigenräume</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E\left(\lambda _{i}\right)}">

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E\left(\lambda _{i}\right)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e84cbe1d33eb741bf5991f5438528bc33ff783c9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.127ex; height:2.843ex;" alt="{\displaystyle E\left(\lambda _{i}\right)}" loading="lazy"> zu allen Eigenwerten <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda _{i}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \lambda _{i}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/72fde940918edf84caf3d406cc7d31949166820f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.155ex; height:2.509ex;" alt="{\displaystyle \lambda _{i}}" loading="lazy"> berechnet, also Gleichungssysteme der folgenden Form gelöst

<msub>

</mrow>

</msub>

<mrow>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (A-\lambda _{i}I)\cdot {\begin{pmatrix}e_{1}\\\vdots \\e_{n}\end{pmatrix}}=0}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2fcfdb675a6b8814405bc5d813cffdc2be6ceeb3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.671ex; width:22.78ex; height:10.509ex;" alt="{\displaystyle (A-\lambda _{i}I)\cdot {\begin{pmatrix}e_{1}\\\vdots \\e_{n}\end{pmatrix}}=0}" loading="lazy">.</dd></dl></li>

<li>Weil die geometrische Vielfachheit gleich der algebraischen Vielfachheit jedes Eigenwerts ist, kann man zu jeder maximalen Menge <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda _{i_{1}}=\ldots =\lambda _{i_{k}}}">

<msub>

<msub>

</mrow>

</msub>

</mrow>

</msub>

<msub>

<msub>

</mrow>

</msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \lambda _{i_{1}}=\ldots =\lambda _{i_{k}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e71e116b3f5d4b968b4556d22ac7db09028a5412.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.921ex; height:2.843ex;" alt="{\displaystyle \lambda _{i_{1}}=\ldots =\lambda _{i_{k}}}" loading="lazy"> übereinstimmender Eigenwerte eine Basis <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left\{b_{i_{1}},\ldots ,b_{i_{k}}\right\}}">

<mrow>

<mrow>

<msub>

<msub>

</mrow>

</msub>

</mrow>

</msub>

<msub>

<msub>

</mrow>

</msub>

</mrow>

</msub>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \left\{b_{i_{1}},\ldots ,b_{i_{k}}\right\}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2a42e0af4bf9c7d254dba4a063e570d57d24f89a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.174ex; height:3.176ex;" alt="{\displaystyle \left\{b_{i_{1}},\ldots ,b_{i_{k}}\right\}}" loading="lazy"> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E\left(\lambda _{i_{1}}\right)=\ldots =E(\lambda _{i_{k}})}">

<mrow>

<msub>

<msub>

</mrow>

</msub>

</mrow>

</msub>

</mrow>

<msub>

<msub>

</mrow>

</msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E\left(\lambda _{i_{1}}\right)=\ldots =E(\lambda _{i_{k}})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3ff6c618c8a75496975cbb9143a28f8269404af5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:22.478ex; height:3.009ex;" alt="{\displaystyle E\left(\lambda _{i_{1}}\right)=\ldots =E(\lambda _{i_{k}})}" loading="lazy"> finden.</li>

<li>Nun ist die Diagonalform <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D_{A}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle D_{A}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b78af4a9d2d8aaec85ea4e6b0f997a31def04311.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.389ex; height:2.509ex;" alt="{\displaystyle D_{A}}" loading="lazy"> der Matrix <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"> bezüglich der Basis <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B=\left\{b_{1},\ldots ,b_{n}\right\}}">

<mrow>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle B=\left\{b_{1},\ldots ,b_{n}\right\}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f40d7201c42f3284faa589a7bcf2ab74d901d98c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.633ex; height:2.843ex;" alt="{\displaystyle B=\left\{b_{1},\ldots ,b_{n}\right\}}" loading="lazy">:

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle D_{A}=\operatorname {diag} (\lambda _{1},\lambda _{2},\dots ,\lambda _{n})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/13ad2b2d842138c36062bbcfd8d0b0b2a4051508.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.166ex; height:2.843ex;" alt="{\displaystyle D_{A}=\operatorname {diag} (\lambda _{1},\lambda _{2},\dots ,\lambda _{n})}" loading="lazy"></dd>

<mrow>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S=\left(b_{1},\ldots ,b_{n}\right)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4eb30a69965b7887d148ec94f04c08477f448fbf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.853ex; height:2.843ex;" alt="{\displaystyle S=\left(b_{1},\ldots ,b_{n}\right)}" loading="lazy"></dd></dl></li></ol>

<div class="mw-heading mw-heading3"><h3 id="Simultane_Diagonalisierung">Simultane Diagonalisierung</h3></div>

Gelegentlich will man auch zwei Matrizen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A,B}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A,B}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/96c3298ea9aa77c226be56a7d8515baaa517b90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.541ex; height:2.509ex;" alt="{\displaystyle A,B}" loading="lazy"> mit derselben Transformation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4611d85173cd3b508e67077d4a1252c9c05abca2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}" loading="lazy"> diagonalisieren. Falls das gelingt, gilt

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S^{-1}AS=D_{1}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/033ea82fff9f6a1e5caf194266fbf4c0b5a11c9c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.174ex; height:3.009ex;" alt="{\displaystyle S^{-1}AS=D_{1}}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{-1}BS=D_{2}}">

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S^{-1}BS=D_{2}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a6742ff0fffb37e3613663187663b4712f634777.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.195ex; height:3.009ex;" alt="{\displaystyle S^{-1}BS=D_{2}}" loading="lazy"> und da <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D_{1}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle D_{1}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e7714158afb63a98e7dbc6e885f70d9141c6923a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.979ex; height:2.509ex;" alt="{\displaystyle D_{1}}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D_{2}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle D_{2}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/41b3839c40bd06e3dfea10798dfab41a905af256.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.979ex; height:2.509ex;" alt="{\displaystyle D_{2}}" loading="lazy"> Diagonalmatrizen sind,

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle D_{1}\cdot D_{2}=D_{2}\cdot D_{1}\Rightarrow B\cdot A=SD_{2}S^{-1}\cdot SD_{1}S^{-1}=SD_{1}D_{2}S^{-1}=A\cdot B}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b6205efede79e50a1463ce9d6d095e38772472f2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:71.307ex; height:3.009ex;" alt="{\displaystyle D_{1}\cdot D_{2}=D_{2}\cdot D_{1}\Rightarrow B\cdot A=SD_{2}S^{-1}\cdot SD_{1}S^{-1}=SD_{1}D_{2}S^{-1}=A\cdot B}" loading="lazy">.</dd></dl>

Also müssen die <a href="Endomorphismus" title="Endomorphismus">Endomorphismen</a> miteinander kommutieren. In der Tat gilt auch die Umkehrung: Kommutieren zwei diagonalisierbare Endomorphismen, so können sie simultan diagonalisiert werden. In der <a href="Quantenmechanik" title="Quantenmechanik">Quantenmechanik</a> gibt es für zwei solche Operatoren dann eine Basis aus gemeinsamen Eigenzuständen.

<div class="mw-heading mw-heading3"><h3 id="Beispiel">Beispiel</h3></div>

Sei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A={\begin{pmatrix}1&0&1\\0&2&0\\1&0&1\end{pmatrix}}\in \mathbb {R} ^{3\times 3}}">

<mrow>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A={\begin{pmatrix}1&0&1\\0&2&0\\1&0&1\end{pmatrix}}\in \mathbb {R} ^{3\times 3}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8838bffeb8a2d7e6564d30e97b0960b69a0f4773.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:25.466ex; height:9.176ex;" alt="{\displaystyle A={\begin{pmatrix}1&0&1\\0&2&0\\1&0&1\end{pmatrix}}\in \mathbb {R} ^{3\times 3}}" loading="lazy"> die zu diagonalisierende Matrix. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"> ist sogar orthogonal diagonalisierbar, da <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"> <a href="Symmetrische_Matrix" title="Symmetrische Matrix">symmetrisch</a> ist, d. h., es gilt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A=A^{\mathsf {T}}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A=A^{\mathsf {T}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a40817313212c3351965305c8a294321e97604e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.936ex; height:2.676ex;" alt="{\displaystyle A=A^{\mathsf {T}}}" loading="lazy">.

Die <a href="Eigenwertproblem" class="mw-redirect" title="Eigenwertproblem">Eigenwerte</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda _{i}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \lambda _{i}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/72fde940918edf84caf3d406cc7d31949166820f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.155ex; height:2.509ex;" alt="{\displaystyle \lambda _{i}}" loading="lazy"> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"> lassen sich durch die Nullstellen des <a href="Charakteristisches_Polynom" title="Charakteristisches Polynom">charakteristischen Polynoms</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi _{A}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \chi _{A}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/49baf1caaa804f2d77bfc7570d102ee4a3cafa26.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.92ex; height:2.009ex;" alt="{\displaystyle \chi _{A}}" loading="lazy"> bestimmen:

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<mrow>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \chi _{A}(\lambda )=\det(\lambda E_{3}-A)=\det {\begin{pmatrix}\lambda -1&0&-1\\0&\lambda -2&0\\-1&0&\lambda -1\end{pmatrix}}=\lambda (\lambda -2)^{2}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/71bd7bab94c9e086b26c0d0524cf565cca345c0a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:67.858ex; height:9.176ex;" alt="{\displaystyle \chi _{A}(\lambda )=\det(\lambda E_{3}-A)=\det {\begin{pmatrix}\lambda -1&0&-1\\0&\lambda -2&0\\-1&0&\lambda -1\end{pmatrix}}=\lambda (\lambda -2)^{2}}" loading="lazy"></dd></dl>

Also <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda _{1}=0,\lambda _{2}=2}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \lambda _{1}=0,\lambda _{2}=2}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0e6fcfe66a728d4d590fe7ca939ddad56a943d7c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.375ex; height:2.509ex;" alt="{\displaystyle \lambda _{1}=0,\lambda _{2}=2}" loading="lazy">. Der Eigenwert 2 hat <a href="Algebraische_Vielfachheit" class="mw-redirect" title="Algebraische Vielfachheit">algebraische Vielfachheit</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{\lambda _{2}}=2}">

<msub>

<msub>

</mrow>

</msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a_{\lambda _{2}}=2}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cc5c63fbc2805d6e1ad35e6ca6da22d580caa71d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.513ex; height:2.843ex;" alt="{\displaystyle a_{\lambda _{2}}=2}" loading="lazy">, da er doppelte <a href="Nullstelle" title="Nullstelle">Nullstelle</a> des charakteristischen Polynoms ist.

Zum Bestimmen der <a href="Eigenraum" title="Eigenraum">Eigenräume</a> setze man die Eigenwerte in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E(\lambda )=\mathrm {Kern} (\lambda E_{3}-A)}">

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E(\lambda )=\mathrm {Kern} (\lambda E_{3}-A)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/21b89ec70237c7651755babce6e8bca1b34c7ac0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.601ex; height:2.843ex;" alt="{\displaystyle E(\lambda )=\mathrm {Kern} (\lambda E_{3}-A)}" loading="lazy"> ein.

Um alle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v\in \mathbb {R} ^{3}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle v\in \mathbb {R} ^{3}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bb67eb863a2a7bd76caf55d9a201d5dd8cac9e13.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.701ex; height:2.676ex;" alt="{\displaystyle v\in \mathbb {R} ^{3}}" loading="lazy"> mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\lambda E_{3}-A)v=0}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (\lambda E_{3}-A)v=0}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/979d1fbd8ce3b23d6b16b70c35b1f3d96c3c7187.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.906ex; height:2.843ex;" alt="{\displaystyle (\lambda E_{3}-A)v=0}" loading="lazy"> zu erhalten, fasst man die erweiterte Koeffizientenmatrix <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{pmatrix}\lambda E_{3}-A\mid 0\end{pmatrix}}}">

<mrow>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{pmatrix}\lambda E_{3}-A\mid 0\end{pmatrix}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/248a53886bfeba52b25629844d107fa5574785d6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.369ex; height:2.843ex;" alt="{\displaystyle {\begin{pmatrix}\lambda E_{3}-A\mid 0\end{pmatrix}}}" loading="lazy"> als <a href="Lineares_Gleichungssystem" title="Lineares Gleichungssystem">lineares Gleichungssystem</a> mit unendlichen Lösungen auf.

Für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda =0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \lambda =0}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/00c4bba30544017fe76932de5a4e25adb5512d95.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.616ex; height:2.176ex;" alt="{\displaystyle \lambda =0}" loading="lazy"> erhält man <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{pmatrix}-A\mid 0\end{pmatrix}}={\begin{pmatrix}{\begin{array}{ccc|c}-1&0&-1&0\\0&-2&0&0\\-1&0&-1&0\end{array}}\end{pmatrix}}}">

<mrow>

<mtr>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

<mrow>

<mtr>

<mtd>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{pmatrix}-A\mid 0\end{pmatrix}}={\begin{pmatrix}{\begin{array}{ccc|c}-1&0&-1&0\\0&-2&0&0\\-1&0&-1&0\end{array}}\end{pmatrix}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4baf1f7e2dd26ee480f09fa7534ae8e2cd8f2540.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.505ex; width:36.78ex; height:10.176ex;" alt="{\displaystyle {\begin{pmatrix}-A\mid 0\end{pmatrix}}={\begin{pmatrix}{\begin{array}{ccc|c}-1&0&-1&0\\0&-2&0&0\\-1&0&-1&0\end{array}}\end{pmatrix}}}" loading="lazy">, mit dem <a href="Gau%C3%9Fsches_Eliminationsverfahren" title="Gaußsches Eliminationsverfahren">gaußschen Eliminationsverfahren</a> erhalten wir <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{pmatrix}{\begin{array}{ccc|c}1&0&1&0\\0&1&0&0\\0&0&0&0\end{array}}\end{pmatrix}}}">

<mrow>

<mtr>

<mtd>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{pmatrix}{\begin{array}{ccc|c}1&0&1&0\\0&1&0&0\\0&0&0&0\end{array}}\end{pmatrix}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/715f71e99028469aa022c609670de54a2e970d38.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.505ex; width:19.045ex; height:10.176ex;" alt="{\displaystyle {\begin{pmatrix}{\begin{array}{ccc|c}1&0&1&0\\0&1&0&0\\0&0&0&0\end{array}}\end{pmatrix}}}" loading="lazy"> und somit als <a href="L%C3%B6sungsmenge" title="Lösungsmenge">Lösungsmenge</a> den Eigenraum:

<msub>

</mrow>

</msub>

<mrow>

<mrow>

<mrow>

<mtr>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

</mrow>

</mrow>

<mrow>

<mtr>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E(\lambda _{1})=\left\{{\begin{pmatrix}\alpha \\0\\-\alpha \end{pmatrix}}:\alpha \in \mathbb {R} \right\}=\operatorname {Lin} {\begin{pmatrix}1\\0\\-1\end{pmatrix}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/45bc3cc85345d8d32e1067c75a1265d3dc0c5c50.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:43.949ex; height:9.176ex;" alt="{\displaystyle E(\lambda _{1})=\left\{{\begin{pmatrix}\alpha \\0\\-\alpha \end{pmatrix}}:\alpha \in \mathbb {R} \right\}=\operatorname {Lin} {\begin{pmatrix}1\\0\\-1\end{pmatrix}}}" loading="lazy">,</dd></dl>

wobei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Lin} }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {Lin} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/668c29329c47fc309e931257749a69d6915dd85e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.392ex; height:2.176ex;" alt="{\displaystyle \operatorname {Lin} }" loading="lazy"> die <a href="Lineare_H%C3%BClle" title="Lineare Hülle">lineare Hülle</a> bezeichnet.

Für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda =2}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \lambda =2}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5b05534f7ee0cfcbac25615feeb699abe09569ae.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.616ex; height:2.176ex;" alt="{\displaystyle \lambda =2}" loading="lazy"> erhält man <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{pmatrix}2\cdot E_{3}-A\mid 0\end{pmatrix}}={\begin{pmatrix}{\begin{array}{ccc|c}1&0&-1&0\\0&0&0&0\\-1&0&1&0\end{array}}\end{pmatrix}}}">

<mrow>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

</mtable>

</mrow>

<mrow>

<mtr>

<mtd>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{pmatrix}2\cdot E_{3}-A\mid 0\end{pmatrix}}={\begin{pmatrix}{\begin{array}{ccc|c}1&0&-1&0\\0&0&0&0\\-1&0&1&0\end{array}}\end{pmatrix}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e2039ddf25075e22f5b6f95575e04efcea232724.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.505ex; width:41.615ex; height:10.176ex;" alt="{\displaystyle {\begin{pmatrix}2\cdot E_{3}-A\mid 0\end{pmatrix}}={\begin{pmatrix}{\begin{array}{ccc|c}1&0&-1&0\\0&0&0&0\\-1&0&1&0\end{array}}\end{pmatrix}}}" loading="lazy">, daraus

<mrow>

<mtr>

<mtd>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{pmatrix}{\begin{array}{ccc|c}1&0&-1&0\\0&0&0&0\\0&0&0&0\end{array}}\end{pmatrix}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/24ff1734c3c9de32d9c9d0f3768e2caa6e587800.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.505ex; width:20.854ex; height:10.176ex;" alt="{\displaystyle {\begin{pmatrix}{\begin{array}{ccc|c}1&0&-1&0\\0&0&0&0\\0&0&0&0\end{array}}\end{pmatrix}}}" loading="lazy"> und somit als Lösungsmenge den Eigenraum:

<msub>

</mrow>

</msub>

<mrow>

<mrow>

<mrow>

<mtr>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

</mrow>

</mrow>

<mrow>

<mrow>

<mrow>

<mtr>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

<mrow>

<mtr>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E(\lambda _{2})=\left\{{\begin{pmatrix}\alpha \\\beta \\\alpha \end{pmatrix}}:\alpha ,\beta \in \mathbb {R} \right\}=\operatorname {Lin} \left\{{\begin{pmatrix}1\\0\\1\end{pmatrix}},{\begin{pmatrix}0\\1\\0\end{pmatrix}}\right\}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9fdcfee7a82c1c8d51c385226ef88b651849c3a2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:53.458ex; height:9.176ex;" alt="{\displaystyle E(\lambda _{2})=\left\{{\begin{pmatrix}\alpha \\\beta \\\alpha \end{pmatrix}}:\alpha ,\beta \in \mathbb {R} \right\}=\operatorname {Lin} \left\{{\begin{pmatrix}1\\0\\1\end{pmatrix}},{\begin{pmatrix}0\\1\\0\end{pmatrix}}\right\}}" loading="lazy">.</dd></dl>

Die Eigenvektoren <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{1}={\begin{pmatrix}1\\0\\-1\end{pmatrix}},v_{2}={\begin{pmatrix}1\\0\\1\end{pmatrix}},v_{3}={\begin{pmatrix}0\\1\\0\end{pmatrix}}}">

<msub>

</mrow>

</msub>

<mrow>

<mtr>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

<msub>

</mrow>

</msub>

<mrow>

<mtr>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

<msub>

</mrow>

</msub>

<mrow>

<mtr>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle v_{1}={\begin{pmatrix}1\\0\\-1\end{pmatrix}},v_{2}={\begin{pmatrix}1\\0\\1\end{pmatrix}},v_{3}={\begin{pmatrix}0\\1\\0\end{pmatrix}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f55389e5d006f01408184cab9b37498bf6d29bbf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:37.659ex; height:9.176ex;" alt="{\displaystyle v_{1}={\begin{pmatrix}1\\0\\-1\end{pmatrix}},v_{2}={\begin{pmatrix}1\\0\\1\end{pmatrix}},v_{3}={\begin{pmatrix}0\\1\\0\end{pmatrix}}}" loading="lazy"> erhält man aus den <a href="Basis_(Vektorraum)" title="Basis (Vektorraum)">Basen</a> der Eigenräume, sie bilden eine Basis von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{3}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{3}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f936ddf584f8f3dd2a0ed08917001b7a404c10b5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{3}}" loading="lazy">.

Wenn man <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{1},v_{2},v_{3}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle v_{1},v_{2},v_{3}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/12e5ac2b8e42965e3d97dbe06d1031361bc310e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.613ex; height:2.009ex;" alt="{\displaystyle v_{1},v_{2},v_{3}}" loading="lazy"> <a href="Einheitsvektor" title="Einheitsvektor">normiert</a>, erhält man mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b_{1}={\frac {1}{\sqrt {2}}}{\begin{pmatrix}1\\0\\-1\end{pmatrix}},b_{2}={\frac {1}{\sqrt {2}}}{\begin{pmatrix}1\\0\\1\end{pmatrix}},b_{3}={\begin{pmatrix}0\\1\\0\end{pmatrix}}}">

<msub>

</mrow>

</msub>

<mfrac>

<msqrt>

</msqrt>

</mfrac>

</mrow>

<mrow>

<mtr>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

<msub>

</mrow>

</msub>

<mfrac>

<msqrt>

</msqrt>

</mfrac>

</mrow>

<mrow>

<mtr>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

<msub>

</mrow>

</msub>

<mrow>

<mtr>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle b_{1}={\frac {1}{\sqrt {2}}}{\begin{pmatrix}1\\0\\-1\end{pmatrix}},b_{2}={\frac {1}{\sqrt {2}}}{\begin{pmatrix}1\\0\\1\end{pmatrix}},b_{3}={\begin{pmatrix}0\\1\\0\end{pmatrix}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0c84495cf5a8b75e3497f4506a08cd3be537ccd4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:45.138ex; height:9.176ex;" alt="{\displaystyle b_{1}={\frac {1}{\sqrt {2}}}{\begin{pmatrix}1\\0\\-1\end{pmatrix}},b_{2}={\frac {1}{\sqrt {2}}}{\begin{pmatrix}1\\0\\1\end{pmatrix}},b_{3}={\begin{pmatrix}0\\1\\0\end{pmatrix}}}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {B}}=\left\{b_{1},b_{2},b_{3}\right\}}">

</mrow>

<mrow>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {B}}=\left\{b_{1},b_{2},b_{3}\right\}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cb7aa3a21207da5e55591f0e6e72cf70ebaa74d4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.19ex; height:2.843ex;" alt="{\displaystyle {\mathcal {B}}=\left\{b_{1},b_{2},b_{3}\right\}}" loading="lazy"> eine <a href="Orthonormalbasis" title="Orthonormalbasis">Orthonormalbasis</a>, da <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"> <a href="Symmetrische_Matrix" title="Symmetrische Matrix">symmetrisch</a> und die Eigenvektoren der <a href="Eigenwertproblem#Spektrum_und_Vielfachheiten" class="mw-redirect" title="Eigenwertproblem">halbeinfachen Eigenwerte</a> orthogonal zueinander sind (in dem Fall <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{2}\perp v_{3}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle v_{2}\perp v_{3}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d56ae9dbb0ed51839d54fda6eb77f5c5c2ddf913.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.462ex; height:2.509ex;" alt="{\displaystyle v_{2}\perp v_{3}}" loading="lazy">).

Es gilt also <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S=\left(b_{1},b_{2},b_{3}\right)={\frac {1}{\sqrt {2}}}{\begin{pmatrix}1&1&0\\0&0&{\sqrt {2}}\\-1&1&0\end{pmatrix}}}">

<mrow>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

<mfrac>

<msqrt>

</msqrt>

</mfrac>

</mrow>

<mrow>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

<msqrt>

</msqrt>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S=\left(b_{1},b_{2},b_{3}\right)={\frac {1}{\sqrt {2}}}{\begin{pmatrix}1&1&0\\0&0&{\sqrt {2}}\\-1&1&0\end{pmatrix}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/92b93c2c9c16c058c3e21afeb8c7339ef93a7c1e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:38.358ex; height:9.509ex;" alt="{\displaystyle S=\left(b_{1},b_{2},b_{3}\right)={\frac {1}{\sqrt {2}}}{\begin{pmatrix}1&1&0\\0&0&{\sqrt {2}}\\-1&1&0\end{pmatrix}}}" loading="lazy">. Daraus erhält man unter der Nutzung der Eigenschaften von Orthonormalbasen die <a href="Inverse_Matrix" title="Inverse Matrix">Inverse</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{-1}=S^{\mathsf {T}}={\frac {1}{\sqrt {2}}}{\begin{pmatrix}1&0&-1\\1&0&1\\0&{\sqrt {2}}&0\end{pmatrix}}}">

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<mfrac>

<msqrt>

</msqrt>

</mfrac>

</mrow>

<mrow>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

<msqrt>

</msqrt>

</mrow>

</mtd>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S^{-1}=S^{\mathsf {T}}={\frac {1}{\sqrt {2}}}{\begin{pmatrix}1&0&-1\\1&0&1\\0&{\sqrt {2}}&0\end{pmatrix}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bde61cc3367e1c21e468bcaf120f8bb2d6681d15.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:33.554ex; height:9.509ex;" alt="{\displaystyle S^{-1}=S^{\mathsf {T}}={\frac {1}{\sqrt {2}}}{\begin{pmatrix}1&0&-1\\1&0&1\\0&{\sqrt {2}}&0\end{pmatrix}}}" loading="lazy">.

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle D_{A}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b78af4a9d2d8aaec85ea4e6b0f997a31def04311.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.389ex; height:2.509ex;" alt="{\displaystyle D_{A}}" loading="lazy"> bestimmt sich durch <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D_{A}=\operatorname {diag} (\lambda _{1},\lambda _{2},\lambda _{3})=\operatorname {diag} (0,2,2)={\begin{pmatrix}0&0&0\\0&2&0\\0&0&2\end{pmatrix}}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<mrow>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle D_{A}=\operatorname {diag} (\lambda _{1},\lambda _{2},\lambda _{3})=\operatorname {diag} (0,2,2)={\begin{pmatrix}0&0&0\\0&2&0\\0&0&2\end{pmatrix}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d0c1f47dd23ee459de19c489a47f480f6c35486f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:52.634ex; height:9.176ex;" alt="{\displaystyle D_{A}=\operatorname {diag} (\lambda _{1},\lambda _{2},\lambda _{3})=\operatorname {diag} (0,2,2)={\begin{pmatrix}0&0&0\\0&2&0\\0&0&2\end{pmatrix}}}" loading="lazy">.

Somit erhält man für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D_{A}=S^{-1}AS}">

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle D_{A}=S^{-1}AS}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/29488cd6eaa1bbdb9755fca5435a6f7f4653df15.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.584ex; height:3.009ex;" alt="{\displaystyle D_{A}=S^{-1}AS}" loading="lazy">

<mrow>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

<mfrac>

</mfrac>

</mrow>

<mrow>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

<msqrt>

</msqrt>

</mrow>

</mtd>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

<mrow>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

<mrow>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

<msqrt>

</msqrt>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{pmatrix}0&0&0\\0&2&0\\0&0&2\end{pmatrix}}={\frac {1}{2}}{\begin{pmatrix}1&0&-1\\1&0&1\\0&{\sqrt {2}}&0\end{pmatrix}}{\begin{pmatrix}1&0&1\\0&2&0\\1&0&1\end{pmatrix}}{\begin{pmatrix}1&1&0\\0&0&{\sqrt {2}}\\-1&1&0\end{pmatrix}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/65f023548f4326371c3c5a2a33e03cb943e55edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:64.389ex; height:9.509ex;" alt="{\displaystyle {\begin{pmatrix}0&0&0\\0&2&0\\0&0&2\end{pmatrix}}={\frac {1}{2}}{\begin{pmatrix}1&0&-1\\1&0&1\\0&{\sqrt {2}}&0\end{pmatrix}}{\begin{pmatrix}1&0&1\\0&2&0\\1&0&1\end{pmatrix}}{\begin{pmatrix}1&1&0\\0&0&{\sqrt {2}}\\-1&1&0\end{pmatrix}}}" loading="lazy"></dd></dl>

und damit die Diagonalisierung

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

<mfrac>

</mfrac>

</mrow>

<mrow>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

<msqrt>

</msqrt>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

<mrow>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

<mrow>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

<msqrt>

</msqrt>

</mrow>

</mtd>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A=SD_{A}S^{-1}={\frac {1}{2}}{\begin{pmatrix}1&1&0\\0&0&{\sqrt {2}}\\-1&1&0\end{pmatrix}}{\begin{pmatrix}0&0&0\\0&2&0\\0&0&2\end{pmatrix}}{\begin{pmatrix}1&0&-1\\1&0&1\\0&{\sqrt {2}}&0\end{pmatrix}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c9a3f657609ed5b4d3fb506ea645b6d466a7f0ce.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:65.022ex; height:9.509ex;" alt="{\displaystyle A=SD_{A}S^{-1}={\frac {1}{2}}{\begin{pmatrix}1&1&0\\0&0&{\sqrt {2}}\\-1&1&0\end{pmatrix}}{\begin{pmatrix}0&0&0\\0&2&0\\0&0&2\end{pmatrix}}{\begin{pmatrix}1&0&-1\\1&0&1\\0&{\sqrt {2}}&0\end{pmatrix}}}" loading="lazy">.</dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Siehe_auch">Siehe auch</h2></div>

<ul><li><a href="Jordansche_Normalform" title="Jordansche Normalform">Jordansche Normalform</a></li>

<li><a href="Frobenius-Normalform" title="Frobenius-Normalform">Frobenius-Normalform</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Weblinks">Weblinks</h2></div>

<div class="sisterproject" style="margin:0.1em 0 0 0;"><div class="noviewer" style="display:inline-block; line-height:10px; min-width:1.6em; text-align:center;" aria-hidden="true" role="presentation"></div><a href="https://de.wikiversity.org/wiki/Kurs:Lineare_Algebra_(Osnabr%C3%BCck_2024-2025)/Teil_I/Vorlesung_22" class="extiw external" title="v:Kurs:Lineare Algebra (Osnabrück 2024-2025)/Teil I/Vorlesung 22">Wikiversity: Vorlesung über diagonalisierbare Abbildungen</a> – Kursmaterialien</div>

<div class="mw-heading mw-heading2"><h2 id="Einzelnachweise">Einzelnachweise</h2></div>

<li id="cite_note-1"><a href="#cite_ref-1">↑</a> <a href="Uwe_Storch" title="Uwe Storch">Uwe Storch</a>, Hartmut Wiebe: Lehrbuch der Mathematik, Band 2: Lineare Algebra. BI-Wissenschafts-Verlag, Mannheim u. a. 1990, ISBN 3-411-14101-8.

</li>

</ol></div><div><div class="zim-footer">

Dieser Artikel wurde von <a class="external text" title="Zuletzt bearbeitet am 2025-12-12" href="https://de.wikipedia.org/wiki/?title=Diagonalisierbare_Matrix&oldid=262363503">Wikipedia</a> herausgegeben. Der Text ist unter <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.de">Creative Commons Attribution-Share Alike 4.0</a> verfügbar, sofern nicht anders angegeben. Für die Mediendateien können zusätzliche Bedingungen gelten.

</div>

</div></div>

</div>

</main>

</div>

</body></html>