Two matrices A and B are defined to be unitarily equivalent if there exists a unitary matrix P such that A = P*BP. If one is given two matrices A and B with no P, how do we know if the matrices are unitarily equivalent? I guess what I'm asking is, is it sufficient to check whether the matrices have the same characteristic polynomial and that one of them is diagonal? What exactely does normality have to do with unitary equivalence? How is it related to similarity and congruence of matrices? Someone suggested to me that I check to see if they are both normal, but I guess I don't see how that fits into the definition. I know that matrices that are unitarily equivalent are also similar (P seems a lot like regular change of basis matrix Q except that it comes from an orthonormal basis instead of regular bases). Anyone know anything I'm missing?