Quaternion MPCEP, CEPMP, and MPCEPMP Generalized Inverses

Abstract

A generalized inverse of a matrix is an inverse in some sense for a wider class of matrices than invertible matrices. Generalized inverses exist for an arbitrary matrix and coincide with a regular inverse for invertible matrices. The most famous generalized inverses are the Moore–Penrose inverse and the Drazin inverse. Recently, new generalized inverses were introduced, namely the core inverse and its generalizations. Among them, there are compositions of the Moore–Penrose and core inverses, MPCEP (or MP–Core–EP) and EPCMP (or EP–Core–MP) inverses. In this chapter, the notions of the MPCEP inverse and CEPMP inverse are expanded to quaternion matrices and introduced new generalized inverses, the right and left MPCEPMP inverses. Direct method of their calculations, that is, their determinantal representations are obtained within the framework of theory of quaternion row-column determinants previously developed by the author. In consequence, these determinantal representations are derived in the case of complex matrices.

Existence Criteria and Related Relation of the gMP Inverse of Matrices

“…The core-EP inverse of A was introduced in [4], and the symbol of this inverse is A † ⃝ ; X is called the core-EP inverse of A if XAX = X, R(X) = R(X * ) = R(A k ), where A ∈ C n×n k and X ∈ C n×n . The MPCEP inverse of an operator was introduced by Chen, Mosić and Xu [5], and this concept was expanded on quaternion matrices by Kyrchei, Mosić and Stanimirović [6,7]. The MPCEP inverse of A is denoted by A †, † ⃝ , which is a matrix (X ∈ C n×n ) such that XAX = X, AX = AA † ⃝ and XA = A † AA † ⃝ A.…”