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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 manydoi:10.1090/s0002-9939-1989-0977922-1 fatcat:qiyomz74png2fmnmmptlxnburi