The general coupled linear matrix equations with conjugate and transpose unknowns over the mixed groups of generalized reflexive and anti-reflexive matrices

Abstract

In this paper, we consider a class of general coupled linear matrix equations over the complex number field. The mentioned coupled linear matrix equations contain the unknown complex matrix groups \(X=(X_1,X_2,\ldots ,X_q)\) and \(Z=(Z_1,Z_2,\ldots ,Z_q)\). The conjugate and transpose of the unknown matrices \(X_i\) and \(Z_i\), \(i\in I[1,q]\), appear in the considered coupled linear matrix equations. An iterative algorithm is presented to determine the unknown matrix groups \(X\) and \(Z\) such that \(X\) and \(Z\) are the groups of the generalized reflexive and anti-reflexive matrices, respectively. The proposed algorithm determines the solvability of the general coupled linear matrix equations over the generalized reflexive and anti-reflexive matrices, automatically. When the general coupled linear matrix equations are consistent over the generalized reflexive and anti-reflexive matrices, it is shown that the algorithm converges within finite number of steps, in the exact arithmetic. In addition, the optimal approximately generalized reflexive and anti-reflexive solution groups to the given arbitrary matrix groups \(\Gamma _x=(\Gamma _{1x},\Gamma _{2x},\ldots ,\Gamma _{qx})\) and \(\Gamma _z=(\Gamma _{1z},\Gamma _{2z},\ldots ,\Gamma _{qz})\) are derived. Finally, some numerical results are given to illustrate the validity of the presented theoretical results and feasibly of the proposed algorithm.