This work determines the zero-dilation index, the largest size of a zero matrix that dilates to a given matrix, for block companion matrices and upper-triangular KMS matrices. Students learn how, under a nonsingularity condition, the index of the block companion matrix is exactly mn/2 for even block count and confined between (m-1)n/2 and (m+1)n/2 for odd count, and how the index of the KMS matrix equals the number of nonnegative eigenvalues of its Hermitian part, connecting dilation indices to the numerical range.
Computes the zero-dilation index d(A), the largest k such that the k-by-k zero matrix dilates to A, for two families of block matrices: block analogues of companion matrices and upper-triangular KMS …