Abstract
We consider the following problem: Given a set of m × n real (or complex) matrices A 1 , … , A N , find an m × m orthogonal (or unitary) matrix P and an n × n orthogonal (or unitary) matrix Q such that P * A 1 Q , … , P * A N Q 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.
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