Math 6610 • Analysis of Numerical Methods I

Singular Value Decomposition. Let \(A \in \mathbb{R}^{m \times n}\), with \(m \ge n\).
\begin{enumerate}

\item
What is the singular value decomposition (SVD) of \(A\)? Does every such matrix
\(A\) have an SVD? Prove that it does or show an example of a matrix \(A\) which
does not have an SVD.


\item
For any \(m \times n\) matrix \(A\) with an SVD and \(\epsilon > 0\), show that there is a full-rank
matrix \(B\) for which \(\|A - B\|_2 < \epsilon\).


\item
For \(m \times n\) matrix \(A\), give orthonormal bases for the subspaces \(\operatorname{range}(A)\), \(\operatorname{null}(A)\),
\(\operatorname{range}(A^*)\), and \(\operatorname{null}(A^*)\).

\end{enumerate}