Math 6610 • Analysis of Numerical Methods I

{Singular Value Decomposition (SVD):}

\partitem{a} Prove the following statement:

\begin{quote}
Singular Value Decomposition: Any matrix \(A\in\C^{m\times n}\) can be factored
as \(A=U\Sigma V^*\), where \(U\in\C^{m\times m}\) and \(V\in\C^{n\times n}\) are
unitary and \(\Sigma\in\R^{m\times n}\) is a rectangular matrix whose only
nonzero entries are non-negative entries on its diagonal.
\end{quote}

\partitem{b} Relate the matrices \(U\), \(V\), and \(\Sigma\) to the four fundamental
subspaces associated with \(A\), that is, the range and null spaces of \(A\) and
\(A^T\).