Lattice properties of the sharp partial order

TL;DR

This paper explores the lattice structure of the sharp partial order on complex matrices with index at most 1, providing conditions for when down-sets form lattices, and characterizing solutions to related matrix equations.

Contribution

It introduces new conditions for down-sets to be lattices, extends previous results to smaller matrix sizes, and characterizes solutions of matrix equations using isomorphisms.

Findings

Down-sets of matrices can form lattices under certain conditions.

The set of matrices with index at most 1 is a lower semilattice only for n=2.

The paper provides a complete description of elements in these lattices.

Abstract

The aim of this paper is to study lattice properties of the sharp partial order for complex matrices having index at most 1. We investigate the down-set of a fixed matrix $B$ under this partial order via isomorphisms with two different partially ordered sets of projectors. These are, respectively, the set of projectors that commute with a certain (nonsingular) block of a Hartwig-Spindelb\"ock decomposition of $B$ and the set of projectors that commute with the Jordan canonical form of that block. Using these isomorphisms, we study the lattice structure of the down-sets and we give properties of them. Necessary and sufficient conditions under which the down-set of B is a lattice were found, in which case we describe its elements completely. We also show that every down-set of $B$ has a distinguished Boolean subalgebra and we give a description of its elements. We characterize the matrices that are above a given matrix in terms of its Jordan canonical form. Mitra (1987) showed that the set of all $n \times n$ complex matrices having index at most 1 with $n\geq 4$ is not a lower semilattice. We extend this result to $n=3$ and prove that it is a lower semilattice with $n=2$. We also answer negatively a conjecture given by Mitra, Bhimasankaram and Malik (2010). As a last application, we characterize solutions of some matrix equations via the established isomorphisms.