File:Eigenvectors.gif
Summary
The transformation matrix <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dcacd8bee0f5c4d9f5e8e3fc2c4932447e0e2aec" class="mwe-math-fallback-image-inline mw-math-element" aria-hidden="true" style="vertical-align: -1.171ex; width:5.144ex; height:3.343ex;" alt="{\displaystyle {\bigl [}{\begin{smallmatrix}2&1\\1&2\end{smallmatrix}}{\bigr ]}}"> preserves the direction of vectors parallel to <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a89dd5be97b160a39b05b9895e4d2a5f760848e8" class="mwe-math-fallback-image-inline mw-math-element" aria-hidden="true" style="vertical-align: -1.171ex; width:3.732ex; height:3.343ex;" alt="{\displaystyle {\bigl (}{\begin{smallmatrix}1\\1\end{smallmatrix}}{\bigr )}}"> (in blue) and <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e26de2a1e3850cffdafa8e9615c0ae665c69e938" class="mwe-math-fallback-image-inline mw-math-element" aria-hidden="true" style="vertical-align: -1.171ex; width:5.017ex; height:3.343ex;" alt="{\displaystyle {\bigl (}{\begin{smallmatrix}1\\-1\end{smallmatrix}}{\bigr )}}"> (in violet). The points that lie on the line through the origin, parallel to an eigenvector, remain on the line after the transformation. The vectors in red are not eigenvectors, therefore their direction is altered by the transformation.
Notice that the blue vectors are scaled by a factor of 3. This is their associated eigenvalue. The violet vectors are not scaled, so their eigenvalue is 1.
Licensing
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File history
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| Date/Time | Thumbnail | Dimensions | User | Comment | |
|---|---|---|---|---|---|
| current | 09:59, 7 January 2017 | | 300 × 300 (58 KB) | 127.0.0.1 (talk) | <p>The transformation matrix <span><span class="mwe-math-mathml-inline mwe-math-mathml-a11y mw-math-element" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0"><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-OPEN"><mo maxsize="1.2em" minsize="1.2em">[</mo></mrow></mrow><mrow class="MJX-TeXAtom-ORD"><mstyle scriptlevel="1"><mtable rowspacing=".2em" columnspacing="0.333em" displaystyle="false"><mtr><mtd><mn>2</mn></mtd><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mn>1</mn></mtd><mtd><mn>2</mn></mtd></mtr></mtable></mstyle></mrow><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-CLOSE"><mo maxsize="1.2em" minsize="1.2em">]</mo></mrow></mrow></mstyle></mrow><annotation encoding="application/x-tex">{\displaystyle {\bigl [}{\begin{smallmatrix}2&1\\1&2\end{smallmatrix}}{\bigr ]}}</annotation></semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dcacd8bee0f5c4d9f5e8e3fc2c4932447e0e2aec" class="mwe-math-fallback-image-inline mw-math-element" aria-hidden="true" style="vertical-align: -1.171ex; width:5.144ex; height:3.343ex;" alt="{\displaystyle {\bigl [}{\begin{smallmatrix}2&1\\1&2\end{smallmatrix}}{\bigr ]}}"></span> preserves the direction of vectors parallel to <span><span class="mwe-math-mathml-inline mwe-math-mathml-a11y mw-math-element" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0"><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-OPEN"><mo maxsize="1.2em" minsize="1.2em">(</mo></mrow></mrow><mrow class="MJX-TeXAtom-ORD"><mstyle scriptlevel="1"><mtable rowspacing=".2em" columnspacing="0.333em" displaystyle="false"><mtr><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mn>1</mn></mtd></mtr></mtable></mstyle></mrow><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-CLOSE"><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow></mrow></mstyle></mrow><annotation encoding="application/x-tex">{\displaystyle {\bigl (}{\begin{smallmatrix}1\\1\end{smallmatrix}}{\bigr )}}</annotation></semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a89dd5be97b160a39b05b9895e4d2a5f760848e8" class="mwe-math-fallback-image-inline mw-math-element" aria-hidden="true" style="vertical-align: -1.171ex; width:3.732ex; height:3.343ex;" alt="{\displaystyle {\bigl (}{\begin{smallmatrix}1\\1\end{smallmatrix}}{\bigr )}}"></span> (in blue) and <span><span class="mwe-math-mathml-inline mwe-math-mathml-a11y mw-math-element" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0"><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-OPEN"><mo maxsize="1.2em" minsize="1.2em">(</mo></mrow></mrow><mrow class="MJX-TeXAtom-ORD"><mstyle scriptlevel="1"><mtable rowspacing=".2em" columnspacing="0.333em" displaystyle="false"><mtr><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mo>−<!-- − --></mo><mn>1</mn></mtd></mtr></mtable></mstyle></mrow><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-CLOSE"><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow></mrow></mstyle></mrow><annotation encoding="application/x-tex">{\displaystyle {\bigl (}{\begin{smallmatrix}1\\-1\end{smallmatrix}}{\bigr )}}</annotation></semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e26de2a1e3850cffdafa8e9615c0ae665c69e938" class="mwe-math-fallback-image-inline mw-math-element" aria-hidden="true" style="vertical-align: -1.171ex; width:5.017ex; height:3.343ex;" alt="{\displaystyle {\bigl (}{\begin{smallmatrix}1\\-1\end{smallmatrix}}{\bigr )}}"></span> (in violet). The points that lie on the line through the origin, parallel to an eigenvector, remain on the line after the transformation. The vectors in red are not eigenvectors, therefore their direction is altered by the transformation. </p> <p>Notice that the blue vectors are scaled by a factor of 3. This is their associated eigenvalue. The violet vectors are not scaled, so their eigenvalue is 1. </p> |
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