Math 6610 • Analysis of Numerical Methods I

{Singular Value Decomposition}
Suppose that \(A\) is a real \(m\times n\) matrix with \(m\ge n\). 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. 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
\(n\times n\) and nonsingular, how would knowing \(A\)'s SVD help you determine the
sensitivity of the solution of a linear system of the following form
\[
  Ax=b?
\]