Zero-dilation Index of S_n-matrix and Companion Matrix

Abstract

The zero-dilation index $d(A)$ of a square matrix $A$ is the largest $k$ for which $A$ is unitarily similar to a matrix of the form ${\scriptsize\left[\begin{array}{cc} 0_k & \ast\\ \ast & \ast\end{array}\right]}$, where $0_k$ denotes the $k$-by-$k$ zero matrix. In this paper, it is shown that if $A$ is an $S_n$-matrix or an $n$-by-$n$ companion matrix, then $d(A)$ is at most $\lceil n/2\rceil$, the smallest integer greater than or equal to $n/2$. Those $A$'s for which the upper bound is attained are also characterized. Among other things, it is shown that, for an odd $n$, the $S_n$-matrix $A$ is such that $d(A)=(n+1)/2$ if and only if $A$ is unitarily similar to $-A$, and, for an even $n$, every $n$-by-$n$ companion matrix $A$ has $d(A)$ equal to $n/2$