Given two unitary and hermitian matrices with equal traces, Prove that they are similar and congruence matrices.

Question

Given two unitary and hermitian matrices with equal traces $A,B$, I'm trying to prove that they are similar and congruence matrices. I proved previously that if these two matrices have equal determinant but not necessarily equal traces, then they aren't necessarily similar congruence, by showing two examples of different cases, but in the case when they have the equal trace, I can't figure a way to show that these two matrices are similar and congruence. Any hint?

Answer 1

Matrix “A” is Hermitian and Unitary. Hence eigenvalues of A are real and lie on the unit circle. Hence eigenvalues of A can be only 1 or -1. Same is the case for matrix B. Since traces of A and B are equal, the arithmetic multiplicities of the eigenvalues 1 and -1 for A and B must be equal individually which means A and B have the same eigenvalues. So they are similar. Moreover A is unitarily diagonalisable and hence it’s congruent to a diagonal matrix D and B is also congruent to the same diagonal matrix D since A and B have the same eigenvalues. Since congruence is an Equivalence Relation we conclude A congruent to B.