Could you show it using the determinate,i.e.,
Let A be an n x n symmetric matrix, and let λ be an eigenvalue of A with corresponding eigenvector x.
Show that λ is also an eigenvalue of the matrix P^T AP where P is an orthogonal matrix. State the corresponding eigenvector of P^T AP.
How is the result modified if P^T AP is diagonal matrix D?
Well, the eigenvalues of a matrix are given by , but
. However, we can prove that by the same reasoning above. Since these have the same solution, i.e., , then the eigenvalues of are the same. Is that a bit clearer? It is not a formal proof, but that should be left up to you to write out.