An algorithm is proposed for finding the finest simultaneous block-diagonalization of a finite number of square matrices, or equivalently the irreducible decomposition of a matrix *-algebra given in terms of its generators. This extends the approach initiated by Murota–Kanno–Kojima–Kojima. The algorithm, composed of numerical-linear algebraic computations, does not require any algebraic structure to be known in advance. The main ingredient of the algorithm is the Schur decomposition and its skew-Hamiltonian variant for eigenvalue computation.
Matrix *-algebra Block-diagonalization Group symmetry Schur decomposition Skew-Hamiltonian Schur decomposition