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

• Fatemeh Panjeh Ali Beik
• Davod Khojasteh Salkuyeh
Original Research

## 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.

### Keywords

Linear matrix equation Iterative algorithm Hermitian reflexive matrix Skew-Hermitian matrix

15A24 65F10

## Notes

### Acknowledgments

The authors would like to express their heartfelt gratitude to the anonymous referees for their valuable suggestions and constructive comments which have improved the quality of the paper.

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