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 in Part I 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.
METR 2008-26, Department of Mathematical Informatics, University of Tokyo, Japan, May 2008 (revised in May 2009).