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<p>If <em>A</em> is symmetric and positive definite, there is an orthogonal matrix <em>Q</em> for which <em>A = Q</em>Λ<em>Q<sup>T</sup></em>. Here Λ is the matrix of eigenvalues. Singular Value Decomposition lets us write <em>any</em> matrix <em>A</em> as a product <em>U</em>Σ<em>V<sup>T</sup></em> where <em>U</em> and <em>V</em> are orthogonal and Σ is a diagonal matrix whose non-zero entries are square roots of the eigenvalues of <em>A<sup>T</sup>A.</em> The columns of <em>U</em> and <em>V</em> give bases for the four fundamental subspaces.</p>
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<h3>Lecture Summary</h3>
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<h3>Suggested Reading</h3>
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<li>Read Section 6.7 in the 4<sup>th</sup> edition or Section 7.1 and 7.2 in the 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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