Factorization of matrices into partial isometries
Kung-Hwang Kuo, Pei Yuan Wu · Proceedings of the American Mathematical Society · 1989
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 T is the product of k k partial isometries ( k ≥ 1 ) (k \geq 1) if and only if T T is a contraction ( ‖ T ‖ ≤ 1 ) (\left \| T \right \| \leq 1) and rank ( 1 − T ∗ T ) ≤ k ⋅ (1 - {T^*}T) \leq k \cdot nullity T T . It follows, as a corollary, that any n × n n \times n singular contraction is the product of n n partial isometries and n n is the smallest such number. On the other hand, T T is the product of finitely many orthogonal projections if and only if T T is unitarily equivalent to 1 ⊕ S 1 \oplus S , where S S