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<p>If the product <em>A</em>x points in the same direction as the vector x, we say that x is an <em>eigenvector</em> of <em>A</em>. Eigenvalues and eigenvectors describe what happens when a matrix is multiplied by a vector. In this session we learn how to find the eigenvalues and eigenvectors of a matrix.</p>
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<h3>Lecture Summary</h3>
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<h3>Suggested Reading</h3>
<ul>
<li>Read Section 6.1 through 6.2 in the 4<sup>th</sup> or 5<sup>th</sup> edition.</li>
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<p>Work the problems on your own and check your answers when you're done.</p>
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<h3>Eigenvalue Demonstrations </h3>
<p>These demonstrations employ Java® applets with voice-over narration by Professor Strang.</p>
<ul>
<li><a href="https://ocw.mit.edu/ans7870/18/18.06/tools/Applets_sound_all/eigen_sound_all.html" target="[object Object]">Eigenvectors of 2 by 2 matrices </a></li>
<li><a href="https://ocw.mit.edu/ans7870/18/18.06/tools/Power_method/power_method_flash.html" target="[object Object]">Powers A<sup><span style="text-decoration: underline;">n</span></sup>V lead toward the top eigenvalue/eigenvector</a></li>
</ul>
<h3>Mini-Lectures on Eigenvalues </h3>
<ul>
<li><a href="https://ocw.mit.edu/ans7870/18/18.06/tools/individual/eigen_lecture_1.html" target="[object Object]">det(A-λI)=0</a></li>
<li><a href="https://ocw.mit.edu/ans7870/18/18.06/tools/individual/eigen_lecture_2.html" target="[object Object]">Eigenvectors and Trace</a></li>
<li><a href="https://ocw.mit.edu/ans7870/18/18.06/tools/individual/eigen_lecture_3.html" target="[object Object]">Powers</a></li>
<li><a href="https://ocw.mit.edu/ans7870/18/18.06/tools/individual/eigen_lecture_4.html" target="[object Object]">Diagnolization</a></li>
<li><a href="https://ocw.mit.edu/ans7870/18/18.06/tools/individual/eigen_lecture_5.html" target="[object Object]">Differential Equations</a></li>
<li><a href="https://ocw.mit.edu/ans7870/18/18.06/tools/individual/eigen_lecture_6.html" target="[object Object]">Symmetry</a></li>
<li><a href="https://ocw.mit.edu/ans7870/18/18.06/tools/individual/eigen_lecture_7.html" target="[object Object]">Positive Definite</a></li>
<li><a href="https://ocw.mit.edu/ans7870/18/18.06/tools/individual/eigen_lecture_8.html" target="[object Object]">SVD</a></li>
</ul>
<p>* Funding for these demonstrations was provided from <a href="http://news.mit.edu/2000/darbeloff-1025" target="[object Object]">The d'Arbeloff Fund for Excellence in MIT Education </a>as part of The d'Arbeloff Interactive Mathematics Project (d'A I M P). </p>
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