Lattice properties of the sharp partial order and other related partial orders for matrices
[EN] 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 the set of pr...
| Authors: | , , , |
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| Format: | article |
| Publication Date: | 2025 |
| Country: | España |
| Institution: | Universitat Politècnica de València (UPV) |
| Repository: | RiuNet. Repositorio Institucional de la Universitat Politécnica de Valéncia |
| Language: | English |
| OAI Identifier: | oai:riunet.upv.es:10251/232569 |
| Online Access: | https://riunet.upv.es/handle/10251/232569 |
| Access Level: | Open access |
| Keyword: | Sharp partial order Hartwig-Spindelböck factorization Lattice structure Jordan canonical form |
| Summary: | [EN] 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 the set of projectors that commute with a certain (nonsingular) block of a HartwigSpindelböck 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 × n complex matrices having index at most 1 when n ¿ 4 is not a lower semilattice. We extend this result to n = 3 and prove that it is a lower semilattice when 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. |
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