Exercise13.3

The matrices \(A\) and \(B\) in \(\Cmm\)are unitarily equivalent if there exists a untiary matrix, \(Q \in \Cmm\) such that \(A=QBQ^\star\text{.}\)

  1. Show that, if \(A\) and \(B\) are unitarily equivalent, then they have the same singular values.
  2. Suppose that \(A, B \in \Cmm\) have the same singular values. Must they be unitarily equivalent?
in-context