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Iterative solutions to the extended Sylvester-conjugate matrix equations. (English) Zbl 1223.65032
The authors investigate the extended Sylvester-conjugate matrix equation $AXB+C\overline{X}D=F$, where $A,C\in {ℂ}^{m×r}$, $B,D\in {ℂ}^{s×n}$ and $F\in {ℂ}^{m×n}$ are known matrices, and $X\in {ℂ}^{r×s}$ is the matrix to be determined. They present an iterative algorithm to solve the extended Sylvester-conjugate matrix equation and a sufficient condition to guarantee that the proposed algorithm converges to the exact solution for arbitrary initial matrix. In addition, a more general Sylvester-conjugate matrix equation of the form ${\sum }_{i=1}^{p}{A}_{i}X{B}_{i}+{\sum }_{j=1}^{q}{C}_{j}\overline{X}{D}_{j}=F$ is also considered in this paper, where ${A}_{i},{C}_{j}\in {ℂ}^{m×r}$, ${B}_{i},{D}_{j}\in {ℂ}^{s×n}\phantom{\rule{4pt}{0ex}}\left(1\le i\le p,\phantom{\rule{4pt}{0ex}}1\le j\le q\right)$ and $F\in {ℂ}^{m×n}$ are known matrices, and $X\in {ℂ}^{r×s}$ is the matrix to be determined, and the convergence analysis is given as well.
##### MSC:
 65F30 Other matrix algorithms 15A24 Matrix equations and identities
##### References:
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