•  
  •  
 

Keywords

Zero-dilation index, Sn-matrix, companion matrix, numerical range.

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 $n/2$

abs_vol31_pp666-678.pdf (32 kB)
Abstract

Included in

Analysis Commons

Share

COinS
 

To view the content in your browser, please download Adobe Reader or, alternately,
you may Download the file to your hard drive.

NOTE: The latest versions of Adobe Reader do not support viewing PDF files within Firefox on Mac OS and if you are using a modern (Intel) Mac, there is no official plugin for viewing PDF files within the browser window.