The $★$-congruence Sylvester equation is the matrix equation $AX+X^{\star }B=C$, where $A\in {\mathbb{F}}^{m\times n}$, $B\in {\mathbb{F}}^{n\times m}$ and $C\in {\mathbb{F}}^{m\times m}$ are given, whereas $X\in {\mathbb{F}}^{n\times m}$ is to be determined. Here, $\mathbb{F}=\mathbb{R}$ or $\mathbb{C}$, and $\star =\mathrm{T}$ (transposed) or $\ast$ (conjugate transposed). Very recently, Satake et al. showed that under some conditions, the matrix equation for the case $\star =\mathrm{T}$ is equivalent to the generalized Sylvester equation. In this paper, we demonstrate that the result can be extended to the case $\star =\ast$. Through this extension, the least squares solution of the $\ast$-congruence Sylvester equation may be obtained using well-researched results on the least squares solution of the generalized Sylvester equation.

的 $★$一致性Sylvester方程是矩阵方程 $\mathrm{一种}X+X^{\star }乙=C$，在哪里 $\mathrm{一种}\in {\mathbb{F}}^{米\times ñ}$， $乙\in {\mathbb{F}}^{ñ\times 米}$ 和 $C\in {\mathbb{F}}^{米\times 米}$ 被给予，而 $X\in {\mathbb{F}}^{ñ\times 米}$待定。这里，$\mathbb{F}=\mathbb{[R}$ 要么 $\mathbb{C}$和 $\star =\mathrm{Ť}$ （转置）或 $\ast$（共轭转置）。最近，Satake等。表明在某些条件下，案例的矩阵方程$\star =\mathrm{Ť}$等价于广义的Sylvester方程。在本文中，我们证明了结果可以扩展到案例$\star =\ast$。通过此扩展，可以求出最小二乘解$\ast$可以使用关于广义Sylvester方程的最小二乘解的深入研究结果来获得-congruence Sylvester方程。