Going through Axler's awful book on linear algebra. The complex spectral theorem (for operator T on vector space V) states that the following are equivalent: 1) T is normal 2) V has an orthonormal basis consisting of eigenvectors of T and 3) the matrix representation of T is diagonal with respect to some orthonormal basis of V. The question I have is: does that imply that T is only diagonalizable if T is normal (on a COMPLEX inner product space)? I.e., is it possible for T to still be diagonalizable with NON-ORTHOGONAL eigenvectors of T if T is not normal? Same question for a real inner product space (R.I.P.S.) and T being not self-adjoint. Now if T is not self-adjoint (on a R.I.P.S.) there is no guarantee that real eigenvalues even exist so I assume that answer must be no.