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Factorization of matrices into partial isometries

Kung Hwang Kuo, Pei Yuan Wu
1989 Proceedings of the American Mathematical Society  
In this paper, we characterize complex square matrices which are expressible as products of partial isometries and orthogonal projections. More precisely, we show that a matrix T is the product of k partial isometries (k > 1) if and only if T is a contraction (||r|| < 1) and rank (1 -T*T) < k • nullity T . It follows, as a corollary, that any n x n singular contraction is the product of n partial isometries and n is the smallest such number. On the other hand, T is the product of finitely many
more » ... t of finitely many orthogonal projections if and only if T is unitarily equivalent to 1 © 5 , where 5 is a singular strict contraction (||5|| < 1) . As contrasted to the previous case, the number of factors can be arbitrarily large.
doi:10.1090/s0002-9939-1989-0977922-1 fatcat:qiyomz74png2fmnmmptlxnburi