Pseudo Schur Complements, Pseudo Principal Pivot Transforms and Their Inheritance Properties

TL;DR

This paper explores properties of pseudo Schur complements and pseudo principal pivot transforms, extending classical matrix concepts by replacing inverses with Moore-Penrose inverses, aiming to facilitate future research applications.

Contribution

It establishes fundamental results on pseudo Schur complements and pseudo principal pivot transforms, extending classical matrix theory with Moore-Penrose inverse-based definitions.

Findings

Basic inheritance properties proved

Extensions of classical concepts established

Results intended for future research use

Abstract

In this short note, we prove some basic results on pseudo Schur complement and the pseudo principal pivot transform of a block matrix. Pseudo Schur complement and pseudo principal pivot ransform are extensions of the Schur complement and the principal pivot transform, respectively, where the usual inverse is replaced by the Moore-Penrose inverse. The objective is to record these results for use in future.