On matrices over an arbitrary semiring and their generalized inverses

Abstract

In this paper, we consider matrices with entries from a semiring S. We first discuss some generalized inverses of rectangular and square matrices. We establish necessary and sufficient conditions for the existence of the Moore–Penrose inverse of a regular matrix. For an m×n matrix A, an n×m matrix P and a square matrix Q of order m, we present necessary and sufficient conditions for the existence of the group inverse of QAP with the additional property that P(QAP)#Q is a {1,2} inverse of A. The matrix product used here is the usual matrix multiplication. The result provides a method for generating elements in the set of {1,2} inverses of an m×n matrix A starting from an initial {1} inverse of A. We also establish a criterion for the existence of the group inverse of a regular square matrix. We then consider a semiring structure (Mm×n(S),+,∘) made up of m×n matrices with the addition defined entry-wise and the multiplication defined as in the case of the Hadamard product of complex matrices. In the semiring (Mm×n(S),+,∘), we present criteria for the existence of the Drazin inverse and the Moore–Penrose inverse of an m×n matrix. When S is commutative, we show that the Hadamard product preserves the Hermitian property, and provide a Schur-type product theorem for the product A∘(CC⁎) of a positive semidefinite n×n matrix A and an n×n matrix C.