A, B normal. Prove that A is similar to B if and only if the characteristic polynomials for A, B are the same.

Question

Suppose A, B ∈ Mn (C) are normal matrices. Prove that A is similar to B if and only if the characteristic polynomials for A, B are the same. (This statement is not true if A, B are not normal matrices.)

Answer 1

If $A$ and $B$ are similar, then they have the same characteristic polynomial, since $det(P(A-xI)P^{-1})=det(A-xI)$.

Suppose that $A$ and $B$ are normal and have the same characteristic polynomial. Since $A$ and $B$ are normal, there exists matrices $P$ and $Q$ such that $PAP^{-1}$ and $QBQ^{-1}$ are diagonal (see the link), since $A$ and $B$ have the same characteristic polynomial, you deduce that $PAP^{-1}=QBQ^{-1}$ thus $A$ and $B$ are similar.

https://en.wikipedia.org/wiki/Normal_matrix#Consequences