One sided Star and Core orthogonality of matrices

TL;DR

This paper explores new types of one-sided orthogonality in matrices, providing foundational results, canonical forms, and conditions for inverse additivity related to these orthogonalities.

Contribution

It introduces and characterizes left and right $*$-orthogonality and core-orthogonality of matrices, linking them to canonical forms and inverse properties.

Findings

Established basic properties and canonical forms for these orthogonalities

Derived conditions for the additivity of Moore-Penrose and core inverses

Explored relations between orthogonal matrices and parallel sums

Abstract

We investigate two one-sided orthogonalities of matrices, the first of which is left (right) $*$-orthogonality for rectangular matrices and the other is left (right) core-orthogonality of index $1$ matrices. We obtain some basic results for these matrices, their canonical forms, and characterizations. Also, relations between left (right) orthogonal matrices and parallel sums are investigated. Finally under these one-sided orthogonalities we explore the conditions of additivity of the Moore-Penrose inverse and the core inverse.