Some results on core EP Drazin matrices and partial isometries

TL;DR

CEPD matrices are defined by the commutation of the core EP inverse and the Drazin inverse, $A^{\textcircled{†}} A^D = A^D A^{\textcircled{†}}$, capturing EP and normal matrices as central examples. The paper develops a framework of equivalent conditions, direct-sum stability, and special cases (e.g., $ind(A)\le 2$) to characterize CEPD, and it analyzes how partial isometries interact with various inverses, providing criteria for when these inverses preserve partial isometry structure or yield CEPD. It also provides practical applications to solving linear systems involving singular or structured matrices and demonstrates results with numerical examples. Overall, the work broadens the understanding of how the Drazin and core EP inverses interact and offers tools for structured inverse-based problem solving in linear algebra.

Abstract

In this paper, by using the core EP inverse and the Drazin inverse which are two well known generalized inverses, a new class of matrices entitled core EP Drazin matrices (shortly, CEPD matrices) is introduced. This class contains the set of all EP matrices and also the set of normal matrices. Some algebraic properties of these matrices are also investigated. Moreover, some results about the Drazin inverse and the core EP inverse of partial isometries are derived, and using them, some conditions for which partial isometries are CEPD, are obtained. To illustrate the main results, some numerical examples are given.

Theorems & Definitions (42)

• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• Definition 3.1
• Theorem 3.2
• proof
• Theorem 3.3
• proof
• Theorem 3.4
• proof