In this article we consider Re-nnd solutions of the equation AX B = C with respect to X , where A, B, C are given matrices. We give necessary and sufficient conditions for the existence of Re-nnd solutions and present a general form of such solutions. As a special case when A = I we obtain the results from a paper of Groß ('Explicit solutions to the matrix inverse problem AX = B', Linear Algebra Appl. 289 (1999), 131-134).2000 Mathematics subject classification: 15A24, 47A62. Keywords and phrases: matrix equation, Hermitian part, Re-nnd solutions.
IntroductionLet C n×m denote the set of complex n × m matrices. Here I n denotes the unit matrix of order n. By A * , R(A), rank(A) and N (A), we denote the conjugate transpose, the range, the rank and the null space of A ∈ C n×m .The Hermitian part of X is defined as H (X ) = (1/2)(X + X * ). We say that X is Re-nnd (Re-nonnegative definite) if H (X ) ≥ 0 and X is Re-pd (Re-positive definite) if H (X ) > 0.The symbol A − stands for an arbitrary generalized inner inverse of A, that is, A − satisfies A A − A = A. By A † we denote the Moore-Penrose inverse of A ∈ C n×m , that is, the unique matrix A † ∈ C m×n satisfyingFor some important properties of generalized inverses see [5,6,16] found the set of all complex Re-nnd matrices X such that X B = C and presented a criterion for Re-nndness. Groß [11] gave an alternative approach, which simultaneously delivers explicit Re-nnd solutions and gave a corrected version of some results from [22]. Some results from [22] were extended in the paper of Wang and Yang [20], in which the authors presented criteria for 2 × 2 and 3 × 3 partitioned matrices to be Re-nnd, found necessary and sufficient conditions for the existence of Re-nnd solutions of the equation (1.1) and derived an expression for these solutions. In the paper of Dajić and Koliha [3], a general form of Re-nnd solutions of the equation AX = C is given for the first time, where A and C are given operators between Hilbert spaces. In addition to these papers many other papers have dealt with the problem of finding the Re-nnd and Re-pd solutions of some other forms of equations.In this paper, we first consider the matrix equationwhere A ∈ C n×m , C ∈ C n×n , and find necessary and sufficient conditions for the existence of Re-nnd solutions. Also, we present a general form of these solutions. Using this result, we obtain necessary and sufficient conditions for the equationwhere A ∈ C n×m , B ∈ C m×n and C ∈ C n×n , to have a Re-nnd solution. This way, the results of [22] and [11] follow as a corollary and a general form of those solutions is given in addition. As far as the author is aware, this is the first time necessary and sufficient conditions for the existence of a Re-nnd solution of the equation AX B = C have been given in terms of g-inverses. Now, we state some well-known results which are used frequently in the next section.
