Characterizations and representations for the {${\mathfrak{m}}$}-core-EP inverse

Abstract

In this paper, we provide some characterizations and representations for the {${\mathfrak{m}}$}-core-EP inverse.
We give a relationship between the ${\mathfrak{m}}$-core-EP inverse and an invertible bordered matrix.
Also, the charactizations for the ${\mathfrak{m}}$-core-EP inverse as an \{2\}-inverse with prescribed range and null space are presented.
The Cramer's rule for the solution of a singular equation $Ax = b$ is also given. The perturbation bounds related with the ${\mathfrak{m}}$-core-EP inverse are estimated.
Furthermore, the successive matrix squaring algorithm for computing the ${\mathfrak{m}}$-core-EP inverse is constructed. Finally,
we show that the ${\mathfrak{m}}$-core-EP inverse can be used in solving appropriate systems of linear equations.