Orthogonal invariants of a matrix of order four and applications

We determine explicitly the algebras of SO4(C)-invariants and O4(C)-invariants of a traceless matrix A of order 4, i.e., we find a set of homogeneous system parameters, minimal set of algebra generators, and Hironaka decomposition for each of these algebras. We have also computed the Hilbert series for the algebra of SOn(C)-invariants of a single matrix A of order n⩽6. All this was originally motivated by the question of orthogonal tridiagonalizability of real matrices of order 4. We show that the answer to this question is negative. It is also negative in the case of complex matrices of order 4 acted upon by the usual complex orthogonal group O4(C).