Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2002 / Paper 1 / 5 / b / Solution

As $C$ is a unitary matrix, $C^{-1}=C^{†}$. Let $M=C^{†}AC$. Its conjugate transpose is $M^{†}=C^{†}{A}^{†}C=M$, so $M$ and every positive integer power $M^n$ are Hermitian matrices and have real eigenvalues.

More explicitly, if $A{v}_j=a_j{v}_j$ with $a_j$ real, then $C^{-1}{v}_j$ form an eigenbasis andThereforewhere $n_A$ denotes the matrix size, to distinguish it from the power $n$.