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

Research paper by Fatemeh Panjeh Ali Beik, Mahmoud Mohseni Moghadam

Indexed on: 16 Nov '13Published on: 16 Nov '13Published in: Computational and Applied Mathematics

#### 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.