Math 6610 • Analysis of Numerical Methods I

{Singular Value Decomposition}
Suppose that \(A\) is a real \(m\times n\) matrix, \(m\ge n\).

\textbf{a)} What is the singular value decomposition (SVD) of \(A\)? Does every
matrix \(A\) of this type have an SVD? If your answer is yes, prove the assertion. If
your answer is no, give an example of a matrix \(A\) which has no SVD.

\textbf{b)} For a matrix \(A\) which does have an SVD, how would having the SVD give
easy access to information about the rank of \(A\)? If \(A\) were \(m\times m\) and
nonsingular, how would knowing \(A\)'s SVD help you determine the sensitivity of the
solution of a linear system of the following form
\[
  Ax=b?
\]

\textbf{c)} For a nonsingular matrix \(A\) which does have an SVD, express the
relative distance from \(A\) to the nearest rank deficient matrix in terms of \(A\)'s
singular values.