Zero-dilation index of S_n-matrix and companion matrix

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 0_k 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 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.