Perforated strict pseudospectra of Demmel's matrices

Abstract

Regarding the pseudospectra of matrices it is easy to derive some conclusions from computer made graphics. However, those conclusions might be false without further analysis. Indeed, it happened when discussing the presence of a local maximum at the origin for the least singular value of the characteristic matrix of the n ×n real Toeplitz matrix whose first row and first column are [−1, −b, −b^2, ..., −b^(n−1)] and [−1, 0, 0, ..., 0]^T, respectively, which was first taken into account by Demmel. In this paper, we determine the value of a local maximum for the last singular value of the characteristic matrix of such Toeplitz matrices in the cases n = 3 and n = 5. We also specify the point where this local maximum is attained at