Determinantal Representations of the Core Inverse and Its Generalizations with Applications

Abstract

In this paper, we give the direct method to find of the core inverse and its generalizations that is based on their determinantal representations. New determinantal representations of the right and left core inverses, the right and left core-EP inverses, and the DMP, MPD, and CMP inverses are derived by using determinantal representations of the Moore-Penrose and Drazin inverses previously obtained by the author. Since the Bott-Duffin inverse has close relation with the core inverse, we give its determinantal representation and its application in finding solutions of the constrained linear equations that is an analog of Cramer’s rule. A numerical example to illustrate the main result is given.

The core inverse and constrained matrix approximation problem

“…and ( ) = some new fields, for example, ring and operator. Moreover, those inverses are used to study partial orders in [4,5,10,18,19].…”

Determinantal Representations of the Weighted Core-EP, DMP, MPD, and CMP Inverses

“…Definition 5 (see [31]). A matrix X ∈ C n×n is said to be the left core inverse of A ∈ C n×n if it satisfies the conditions:…”

“…So, it is enough to investigate the left core-EP inverse, so the right core-EP inverse case can be investigated analogously. But in [31], separate determinantal representations of both core-EP inverses and the both core inverses are given.…”

“…e characterizations, computing methods, some applications of the core inverse, and its generalizations were investigated (see, e.g., [20][21][22][23][24][25][26][27][28][29][30]). Recently, determinantal representations of the core inverse and its generalizations have been obtained in both cases for matrices over the field of complex numbers [31] by using usual determinants and over the quaternion skew field [32] by using noncommutative row-column determinants introduced in [33,34].…”

On forward-order law for core inverse in rings

“…In application point of view, the core inverse is used to find the Bott-Duffin inverse [8]. Also, one can compute the Moore-Penrose inverse and the group inverse by using the forward-order law for the core inverse.…”