Generalized Principal Pivot Transform and its Inheritance Properties

TL;DR

This paper explores advanced properties of the generalized principal pivot transform, including conditions for inverse equality, rank preservation, and inheritance properties, extending existing theoretical frameworks in matrix analysis.

Contribution

It introduces new generalized properties and inheritance conditions for the principal pivot transform, broadening the theoretical understanding beyond previous results.

Findings

Conditions for Moore-Penrose inverse equality are established

Rank preservation of symmetric parts is demonstrated

Inheritance properties of P-matrices are characterized

Abstract

In this paper, some more properties of the generalized principal pivot transform are derived. Necessary and sufficient conditions for the equality between Moore-Penrose inverse of a generalized principal pivot transform and its complementary generalized principal pivot transform are presented. It has been shown that the generalized principal pivot transform preserves the rank of symmetric part of a given square matrix. These results appear to be more generalized than the existing ones. Inheritance property of $P_{\dagger}$-matrix are also characterized for generalized principal pivot transform.