Math 6610 • Analysis of Numerical Methods I

{Unitary matrices}

Let \(A,B\in\C^{N\times N}\). \(A\) and \(B\) are unitarily equivalent if there is a
unitary matrix \(Q\) such that \(A=QBQ^*\). For each of the following statements,
prove that it is true, or demonstrate that it is false:

\partitem{a} If \(A\) and \(B\) are unitarily equivalent, then they have the same
singular values.

\partitem{b} If \(A\) and \(B\) have the same singular values, then they are
unitarily equivalent.