Vanishing Pseudo Schur Complements, Reverse Order Laws, Absorption Laws and Inheritance Properties

TL;DR

This paper investigates conditions under which the vanishing of one generalized Schur complement implies the vanishing of another, explores absorption laws for generalized inverses, and studies inheritance properties related to these concepts.

Contribution

It provides new insights and simple proofs regarding the implications of vanishing Schur complements and their inheritance properties in matrix analysis.

Findings

Simple proof for vanishing Schur complement implications

Derived inheritance properties of generalized Schur complements

Studied inheritance by generalized principal pivot transform

Abstract

The problem of when the vanishing of a (generalized) Schur complement of a block matrix (corresponding to the leading principal subblock) implies that the other (generalized) Schur complement (corresponding to the trailing principal subblock) is zero, is revisited. A simple proof is presented. Absorption laws for two important classes of generalized inverses are considered next. Inheritance properties of the generalized Schur compements in relation to the absorption laws are derived. Inheritance by the generalized principal pivot transform is also studied.