A finite iterative algorithm for Hermitian reflexive and skew-Hermitian solution groups of the general coupled linear matrix equations

Abstract

In this paper, we focus on the following coupled linear matrix equations

$$\begin{aligned} \mathcal {M}_i(X,Y)={\mathcal {M}_{i1}(X)+\mathcal {M}_{i2}(Y)}=L_i, \end{aligned}$$

with

$$\begin{aligned} {\mathcal {M}_{i \ell }(W)}&= \sum \limits _{j = 1}^{q } \left( {{\sum \limits _{\lambda = 1}^{t_1^{(\ell )} } {A_{ij\lambda }^{(\ell )} } } W_j B^{(\ell )}_{ij\lambda } + {\sum \limits _{\mu = 1}^{t_2^{(\ell )} } {C_{ij\mu }^{(\ell )} \overline{W} _j D^{(\ell )}_{ij\mu } } } + {\sum \limits _{\nu = 1}^{t_3^{(\ell )} } {E^{(\ell )}_{ij\nu } W_{j}^T F^{(\ell )}_{ij\nu } } }}\right) , \\&\ell =1,2. \end{aligned}$$

where $A^{(\ell )}_{ij\lambda },B^{(\ell )}_{ij\lambda }$, $C^{(\ell )}_{ij\mu }, D^{(\ell )}_{ij\mu }$, $E^{(\ell )}_{ij\nu },F^{(\ell )}_{ij\nu }$ and $L_i$ (for $i \in I[1,p]$) are given matrices with appropriate dimensions defined over complex number field. Our object is to obtain the solution groups $X=(X_1,X_2,\ldots ,X_q)$ and $Y=(Y_1,Y_2,\ldots ,Y_q)$ of the considered coupled linear matrix equations such that $X$ and $Y$ are the groups of the Hermitian reflexive and skew-Hermitian matrices, respectively. To do so, an iterative algorithm is proposed which stops within finite number of steps in the exact arithmetic. Moreover, the algorithm determines the solvability of the mentioned coupled linear matrix equations over the Hermitian reflexive and skew-Hermitian matrices, automatically. In the case that the coupled linear matrix equations are consistent, the least-norm Hermitian reflexive and skew-Hermitian solution groups can be computed by choosing suitable initial iterative matrix groups. In addition, the unique optimal approximate Hermitian reflexive and skew-Hermitian solution groups to given arbitrary matrix groups are derived. Finally, some numerical experiments are reported to illustrate the validity of our established theoretical results and feasibly of the presented algorithm.