Simultaneous singular value decomposition

We consider the following problem: Given a set of m×n real (or complex) matrices A1,…,AN, find an m×m orthogonal (or unitary) matrix P and an n×n orthogonal (or unitary) matrix Q such that P*A1Q,…,P*ANQ are in a common block-diagonal form with possibly rectangular diagonal blocks. We call this the simultaneous singular value decomposition (simultaneous SVD). The name is motivated by the fact that the special case with N=1, where a single matrix is given, reduces to the ordinary SVD. With the aid of the theory of *-algebra and bimodule it is shown that a finest simultaneous SVD is uniquely determined. An algorithm is proposed for finding the finest simultaneous SVD on the basis of recent algorithms of Murota–Kanno–Kojima–Kojima and Maehara–Murota for simultaneous block-diagonalization of square matrices under orthogonal (or unitary) similarity.