Tridiagonal forms in low dimensions

Pati showed that every 4 × 4 matrix is unitarily similar to a tridiagonal matrix. We give a simple proof. In addition, we show that (in an appropriate sense) there are generically precisely 12 ways to do this. When the real part is diagonal, it is shown that the unitary can be chosen with the form U = PD where D is diagonal and P is real orthogonal. However even if both real and imaginary parts are real symmetric, there may be no real orthogonal matrices which tridiagonalize it. On the other hand, if the matrix belongs to the Lie algebra sp4(C), then it can be tridiagonalized by a unitary in the symplectic group Sp(2). In dimension 5 or greater, there are always rank three matrices which are not tridiagonalizable.