Finite iterative algorithms for the generalized reflexive and anti-reflexive solutions of the linear matrix equation AXB = C

In this paper, an iterative method is presented to solve the linear matrix equation $AXB = C$ over the generalized reflexive (or anti-reflexive) matrix $X$ ($A\in R^{p\times n}, B\in R^{m\times q}, C\in R^{p\times q}, X\in R^{n\times m}$). By the iterative method, the solvability of the equation $AXB = C$ over the generalized reflexive (or anti-reflexive) matrix can be determined automatically. When the equation $AXB = C$ is consistent over the generalized reflexive (or anti-reflexive) matrix $X$, for any generalized reflexive (or anti-reflexive) initial iterative matrix $X_1$ , the generalized reflexive (anti-reflexive) solution can be obtained within finite iterative steps in the absence of roundoff errors. The unique least-norm generalized reflexive (or anti-reflexive) iterative solution of $AXB = C$ can be derived when an appropriate initial iterative matrix is chosen. A sufficient and necessary condition for whether the equation $AXB = C$ is inconsistent is given. Furthermore, the optimal approximate solution of $AXB = C$ for a given matrix $X_0$ can be derived by finding the least-norm generalized reflexive (or anti-reflexive) solution of a new corresponding matrix equation $A\overline{X}B = \overline{C}$. Finally, several numerical examples are given to support the theoretical results of this paper.