$\def\abs#1{|#1|}\def\i{\mathbf {i}}\def\ket#1{|{#1}\rangle}\def\bra#1{\langle{#1}|}\def\braket#1#2{\langle{#1}|{#2}\rangle}\def\tr{\mathord{\mbox{tr}}}\mathbf{Exercise\ 10.9}$

*Singular Value Decomposition.*

Let $A$ be an $n\times m$ matrix.

a) Let $\ket{u_j}$ be unit length eigenvectors of $A^\dagger A$ with eigenvalues $\lambda_j$. Explain how we know that $\lambda_j$ is real and non-negative for all $j$.

b) Let $U$ be the matrix with $\ket{u_j}$ as its columns. Show that $U$ is unitary.

c) For all eigenvectors with non-zero eigenvalues define $\ket{v_i} = \frac{A\ket{x_i} }{\sqrt{\lambda_i} }$. Let $V$ be the matrix with $\ket{v_i}$ as columns. Show that $V$ is unitary.

d) Show that $V^\dagger A U$ is diagonal.

e) Conclude that $A = V D U^\dagger$ for some diagonal $D$. What is $D$?