Chapter 8 - Almost-Hermitian Matrices; Special Three-Row Matrices

Summary

INTRODUCTION

Several authors have tried to study classes of matrices, of modest generality, that contain the matrices (1.3.7) as special cases. The hope was that some insight into the general problem might be gained by deciding the regularity problem for this class. The failure to find a complete solution for even a relatively simple class of this type illustrates the complexity of the general problem, and to some extent justifies its probabilistic reformulation in §6.4.

A matrix E is almost Hermitian if all of its rows are Hermitian except for one interior row. A subclass consists of the three-row matrices E(p, q; k₁, …, k_s) where p and q are the lengths of the Hermitian first and third row while 0 < k₁ < … < k_s are positions of 1's in the second row. We have then p + q + s = n + 1, and we shall always assume 1 ≤ p ≤ q. The regularity problem is not completely solved even for matrices E(p, q; k₁, k₂).

For almost-Hermitian matrices, many useful results follow from a representation of D(E, X) given by Theorem 8.1. Most notable are a strong singularity criterion due to Lorentz [91] and DeVore, Meir, and Sharma [26] in §8.2 and the reciprocity theorem of Drols [29] in §8.3.

Sections 8.4–8.6 deal with the special matrices E(p, q; k₁, k₂). In §8.4 we give the “chasing method” of Lorentz, Stangler, and Zeller [104] in a more geometric form. This solves the problem except for the two cases, (8.2.11) and (8.2.12). The second of them admits a complete solution.