Given a symmetric matrix M with real entries, there exists an orthogonal matrix P such that P^TMP is diagonal. Equivalently, M is diagonalizable with respect to an orthonormal basis. This result generalizes to a Hermitian matrix M when U*MU is real and diagonal for some unitary matrix U and to a self-adjoint linear map of a finite-dimensional inner product space which is likewise diagonalizable with respect to an orthonormal basis. The result is of importance as Hessian, covariance, and inertia matrices are symmetric, and symmetric matrices can also be associated with quadratic forms. Under further technical restrictions, versions of the spectral theorem apply for infinite-dimensional Hilbert spaces.