Simultaneous block diagonalization of matrices of finite order

Abstract It is well known that a set of non-defect matrices can be simultaneously diagonalized if and only the commute. In case non-commuting matrices, best achieved simultaneous block diagonalization. Here we give an efficient algorithm to explicitly compute transfer matrix which realizes diagonalization unitary whose decomposition in irreducible blocks (common invariant subspaces) from elsewhere. Our main motivation lies particle physics, where resulting must order unequivocally determine action outer automorphisms such as parity, charge conjugation, or time reversal on spectrum.