On matrices over an arbitrary semiring and their generalized inverses

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$m\times n$ matrix A  , an n×m$n\times m$ 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$P{(QAP)}^{\mathrm{\#}}Q$ is a {1,2}$\{1,2\}$ 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}$\{1,2\}$ inverses of an m×n$m\times n$ 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),+,°)$({\mathbf{M}}_{m\times n}(\mathbf{S}),+,°)$ made up of m×n$m\times n$ 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),+,°)$({\mathbf{M}}_{m\times n}(\mathbf{S}),+,°)$, we present criteria for the existence of the Drazin inverse and the Moore–Penrose inverse of an m×n$m\times n$ 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^?)$A°(C{C}^{?})$ of a positive semidefinite n×n$n\times n$ matrix A   and an n×n$n\times n$ matrix C.