<p>The transformation matrix <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bigl [}{\begin{smallmatrix}2&amp;1\\1&amp;2\end{smallmatrix}}{\bigr ]}}"><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&amp;1\\1&amp;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" aria-hidden="true" style="vertical-align: -0.919ex; margin-bottom: -0.253ex; width:5.108ex; height:3.343ex;" alt="{\displaystyle {\bigl [}{\begin{smallmatrix}2&amp;1\\1&amp;2\end{smallmatrix}}{\bigr ]}}"></span> preserves the direction of vectors parallel to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bigl (}{\begin{smallmatrix}1\\1\end{smallmatrix}}{\bigr )}}"><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" aria-hidden="true" style="vertical-align: -0.919ex; margin-bottom: -0.253ex; width:3.703ex; height:3.343ex;" alt="{\displaystyle {\bigl (}{\begin{smallmatrix}1\\1\end{smallmatrix}}{\bigr )}}"></span> (in blue) and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bigl (}{\begin{smallmatrix}1\\-1\end{smallmatrix}}{\bigr )}}"><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" aria-hidden="true" style="vertical-align: -1.171ex; width:4.982ex; 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>
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.
<a href="//commons.wikimedia.org/wiki/User:LucasVB" title="User:LucasVB">Lucas Vieira</a>
2012-05-15
<span class="int-own-work" lang="en">Own work</span>