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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2024 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Factorization of matrices into partial isometries
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by Kung Hwang Kuo and Pei Yuan Wu
Proc. Amer. Math. Soc. 105 (1989), 263-272
DOI: https://doi.org/10.1090/S0002-9939-1989-0977922-1

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 $(\left \| T \right \| \leq 1)$ 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 $(\left \| S \right \| < 1)$. As contrasted to the previous case, the number of factors can be arbitrarily large.
References
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Bibliographic Information
  • © Copyright 1989 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 105 (1989), 263-272
  • MSC: Primary 15A23; Secondary 47A68
  • DOI: https://doi.org/10.1090/S0002-9939-1989-0977922-1
  • MathSciNet review: 977922