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====== IAM 961 HW2, fall 2015 ====== **Problem 1:** Prove that any real-valued matrix $A$ has a real-valued SVD, i.e. an SVD factorization $A=U \Sigma V^\dagger$ where $U,\Sigma,$ and $V$ are real-valued matrices. **Problem 2:** Show that if $A$ is m x n and $B$ is p x n, the product $AB^*$ can be written as \begin{eqnarray*} AB^* = \sum_{j=1}^n a_j b^*_j \end{eqnarray*} where $a_j$ and $b_j$ are the columns of $A$ and $B$. This is more a matter of understanding and proper use 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 $x$, <code> A = -0.0954915028125262 -1.2449491424413903 0.6571638901489170 0.7135254915624212 x = 1.50000000000000 1.00000000000000 </code> (a) compute the SVD $A=U \Sigma V^\dagger$. (b) make two plots, one showing the columns of $v_1, v_2$ of $V$ as an orthogonal basis for the domain of $A$, and another showing the columns of $u_1, u_2$ of $U$ as the same for the range of $A$. (c) superimpose a unit circle $C = \{x \vert |x| = 1\}$ on the $v_1, v_2$ plot. (d) superimpose the image of that unit circle under $A$ on the $u_1, u_2$ plot. I.e. for $x \in C$ plot $Ax$. What can you say about the image of $C$ under $A$ in relation to the SVD? (e) Plot the vector $x$ on the $v_1, v_2$ <code> x = 1.50000000000000 1.00000000000000 </code> Figure out where $Ax$ will be on the $u_1, u_2$ plot using the SVD, as follows \begin{eqnarray*} Ax = U \Sigma V^\dagger x = u_1 \sigma_1 v_1^\dagger x + u_2 \sigma_2 v_2^\dagger x \end{eqnarray*}
