Math 6610 • Analysis of Numerical Methods I

(Properties via SVD).
\begin{enumerate}

\item
Consider \(A \in \mathbb{C}^{m \times n}\). Define what we mean by the singular value decomposition of \(A\).


\item
Show that the rank of \(A\) is \(r\), the number of nonzero singular values.


\item
Show that the largest singular value of \(A\), call it here \(\sigma_{\max}(A)\), satisfies the relation
\[
\sigma_{\max}(A) =
\max_{y \in \mathbb{C}^m,\ x \in \mathbb{C}^n}
\frac{|y^*Ax|}{\|x\|_2\|y\|_2}.
\]


\end{enumerate}