A matrix $Pin{mathbb C}^{nimes n}$ is called a generalized reflection matrix if $P^{st}=P$ and $P^{2}=I$. An $nimes n$ complex matrix $A$ is said to be a reflexive (anti-reflexive) matrix with respect to the generalized reflection matrix $P$ if $A=PAP$ ($A=-PAP$). It is well-known that the reflexive and anti-reflexive matrices with respect to the generalized reflection matrix $P$ have many special properties and widely used in engineering and scientific computations. In this paper, we give new necessary and sufficient conditions for the existence of the reflexive (anti-reflexive) solutions to the linear matrix equation $AXB+CYD=E$ and derive representation of the general reflexive (anti-reflexive) solutions to this matrix equation. By using the obtained results, we investigate the reflexive (anti-reflexive) solutions of some special cases of this matrix equation.
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机译：如果$ P ^ { ast} = P $并且$ P ^ {2} = I $，则矩阵$ P in { mathbb C} ^ {n n n} $被称为广义反射矩阵。如果$ A = PAP $（$ A = -PAP $），则相对于广义反射矩阵$ P $，一个$ n×n $的复杂矩阵$ A $被称为是一个自反（反自反）矩阵。众所周知，关于广义反射矩阵$ P $的自反矩阵和反自反矩阵具有许多特殊性质，并广泛用于工程和科学计算中。在本文中，我们为线性矩阵方程$ AXB + CYD = E $的自反（反自反）解的存在给出了新的充要条件，并为此导出了一般自反（反自反）解的表示。矩阵方程。通过使用获得的结果，我们研究了此矩阵方程某些特殊情况的自反（反自反）解。
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