An Efficient Algorithm for the Reflexive Solution of the Quaternion Matrix Equation A X B + C X H D = F

Abstract

We propose an iterative algorithm for solving the reflexive solution of the quaternion matrix equation $AXB+C{X}^{H}D=F$. When the matrix equation is consistent over reflexive matrix $X$, a reflexive solution can be obtained within finite iteration steps in the absence of roundoff errors. By the proposed iterative algorithm, the least Frobenius norm reflexive solution of the matrix equation can be derived when an appropriate initial iterative matrix is chosen. Furthermore, the optimal approximate reflexive solution to a given reflexive matrix $X_0$ can be derived by finding the least Frobenius norm reflexive solution of a new corresponding quaternion matrix equation. Finally, two numerical examples are given to illustrate the efficiency of the proposed methods.