I honorstly don't know... I don't think the fact that two matrices P and Q sum up to a diagonalmatrix D implies that they are diagonalmatrices themselves - it just means that their non-diagonal elements cancel - as you say yourself... Or what?
Let A and B be Hermitian matrices with AB = BA and let N = A + iB.
1) Show that N is normal.
2) Show that A = 1/2(N+N*) (* = conjugate transpose) and find a formula for B.
3) Let U be a unitary matrix such that U*NU is a diagonal matrix. Show that U*AU and U*BU is diagonal matrices...