Factorization of matrices into partial isometries

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.