Hermitian matrices have unitary real diagonalizations; real symmetric matrices have orthogonal diagonalizations.Suggest a slogan
If \(A\in\Mat_n(\CC)\) is Hermitian, that is, \(A^{\dagger} = A\), then \(A\) is diagonalizable over \(\CC\); more precisely, there is a unitary matrix \(U\) such that \(U^{\dagger}AU\) is diagonal with real entries.
If \(A\in\Mat_n(\RR)\) is symmetric, that is, \(A^t = A\), then \(A\) is diagonalizable over \(\RR\); more precisely, there is an orthogonal matrix \(Q\) such that \(Q^tAQ\) is diagonal.