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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Factorization of matrices into partial isometries


Authors: Kung Hwang Kuo and Pei Yuan Wu
Journal: Proc. Amer. Math. Soc. 105 (1989), 263-272
MSC: Primary 15A23; Secondary 47A68
MathSciNet review: 977922
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Abstract: 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 \geq 1)$ if and only if $ T$ is a contraction $ (\left\Vert T \right\Vert \leq 1)$ and rank $ (1 - {T^*}T) \leq k \cdot $ nullity $ T$. It follows, as a corollary, that any $ n \times 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 orthogonal projections if and only if $ T$ is unitarily equivalent to $ 1 \oplus S$, where $ S$ is a singular strict contraction $ (\left\Vert S \right\Vert < 1)$. As contrasted to the previous case, the number of factors can be arbitrarily large.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1989-0977922-1
PII: S 0002-9939(1989)0977922-1
Keywords: Partial isometry, orthogonal projection, contraction, idempotent matrix
Article copyright: © Copyright 1989 American Mathematical Society