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--- gibson:teaching:fall-2015:iam961:iam961-hw2 [2015/09/29 13:38]
gibson
+++ gibson:teaching:fall-2015:iam961:iam961-hw2 [2015/09/30 08:58] (current)
gibson
@@ Line 13: / Line 13: @@
 of notation than any kind of deep proof. It should take about three lines.
-**Problem 3:** Given the following 2 x 2 matrix $A$ and 2-vector $x$,
+**Problem 3:** Let $A$ be an m x n matrix with SVD $A = U\ \Sigma V$. By applying the results of problem 2 to the matrices $U$ and $\Sigma V^*$, show that
+\begin{eqnarray*}
+A = \sum_{j=1}^p \sigma_j u_j v^*_j
+\end{eqnarray*}
+where $p = \min\{m,n\}$. For simplicity you can assume $m \geq n$ so that $p=n$. Note that if $r \leq p$ is the number of nonzero singular values (or equivalently, the rank of $A$), then clearly one can also write the sum going to just $r$ instead of $p$.
+**Problem 4:** Continuing from problem 3, let
+\begin{eqnarray*}
+A_{\nu} = \sum_{j=1}^{\nu} \sigma_j u_j v^*_j
+\end{eqnarray*}
+where $\nu \leq p = \min\{m,n\}$. Show that $A_{\nu}$ is the closest rank-$\nu$ approximation to $A$ in the 2-norm, i.e. that
+\begin{eqnarray*}
+\| A - A_{\nu} \|_2 = \inf_{ B \in \mathbb{C}^{m\times n}, rank{B} \leq \nu}  \| A-B \|_2 = \sigma_{\nu+1}
+\end{eqnarray*}
+(where $\sigma_{\nu+1} = 0$ if $\nu = p$). This is Theorem 5.8 in Trefethen & Bau. You can follow that proof, just write it out in your own words, improving on presentation & argument where you can.
+**Problem 5:** Given the following 2 x 2 matrix $A$ and 2-vector $x$,
 <code>
 A =
@@ Line 36: / Line 54: @@
 .00000000000000
 </code>
-Now figure out where $y=Ax$ will be on the $u_1, u_2$ plot **geometrically* using the SVD, as follows
+Now figure out where $y=Ax$ will be on the $u_1, u_2$ plot **geometrically** using the SVD, as follows
 \begin{eqnarray*}
 y = Ax = U \Sigma V^\dagger x = u_1 \sigma_1 v_1^\dagger x + u_2 \sigma_2 v_2^\dagger x

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