Relative EP matrices

TL;DR

This paper introduces the concept of relative EP matrices with respect to a partial isometry, extending known properties of EP matrices and providing canonical forms and characterizations for rectangular and square matrices.

Contribution

It defines the new concept of T-EP matrices, extends basic EP matrix results, and provides canonical forms and characterizations for these matrices.

Findings

Characterization of T-EP matrices via EP matrices and partial isometries

Canonical forms for rectangular and square matrices

Necessary and sufficient conditions for T-EP matrices

Abstract

The purpose of the present work is to introduce the concept of relative EP matrix of a rectangular matrix relative to a partial isometry (or, in short, $T$-EP matrix) hitherto unknown. We extend various basic results on EP matrices and we study the relationship between $T$-hermitian, $T$-normal and $T$-EP matrices. The main theorems of this paper consist in providing canonical forms of relative EP matrices when matrices involved are rectangular as well as square. We then use them to characterize the relative EP matrices and show their properties. In fact, an interesting fact that has emerged is that $A$ is $T$-EP if and only if there is an EP matrix $C$ such that $A=CT$ and $C=TT^*C$ whatever be the matrix, square or rectangular. We also give various necessary and sufficient conditions for a matrix to be $T$-EP.