General determinantal representation of generalized inverses of matrices over integral domains

Abstract

In this paper we derive a determinantal formula of {1, 2} generalized inverses, for matrices over an integral domain and over a commutative ring. The corresponding results are derived for the set of matrices which have rank factorizations as well as for the matrices which do not have rank factorizations. The determinantal formula of {1, 2} inverses for matrices which do not have rank factorizations, is derived using the characterizations of the class of reflexive g-inverses from [10] and [19]. For the set of matrices which have rank factorizations, the determinantal formula of {1, 2} inverses is derived using a general representation of {1, 2} inverses and the general determinantal representation from [20]. Also, we examine the existence of this determinantal formula. Representations and conditions for the existence of {1, 2, 3} and {1, 2, 4} inverses are introduced for the set of matrices which allow a rank factorization. Determinantal representations of the Moore-Penrose inverse, the weighted Moore-Penrose inverse and the group inverse are derived for arbitrary matrices. Moreover, we investigate representations of the minors from A (1,2) , A † , A † M,N and A (1,2) by means of the expressions involving minors of A and the corresponding minors of randomly chosen matrices which satisfy specified conditions. If A allows a full-rank factorization, we obtain additional results for {1, 2, 3} and {1, 2, 4} inverses of A. Also, a determinantal representation of the corresponding solutions of a given linear system is investigated.

Solving and Algorithm for Least-Norm General Solution to Constrained Sylvester Matrix Equation

“…Due to looking for their more applicable explicit expressions, there are various determinantal representations of generalized inverses (for the MP-inverse, see, e.g. [41,42]). Because of the complexity of the previously obtained expressions of determinantal representations of the MP-inverse, they have little applicability.…”