Zero-dilation index of Sn-matrix and companion matrix

Hwa Long Gau, Pei Yuan Wu

Research output: Contribution to journalArticlepeer-review


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 (Formula presented), where 0k denotes the k-by-k zero matrix. In this paper, it is shown that if A is an Sn-matrix or an n-by-n companion matrix, then d(A) is at most [n/2], 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 Sn-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.

Original languageEnglish
Article number47
Pages (from-to)666-678
Number of pages13
JournalElectronic Journal of Linear Algebra
Issue number1
StatePublished - 2016


  • Companion matrix
  • Numerical range
  • S-matrix
  • Zero-dilation index


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