中文摘要：

考虑矩阵方程组{A1XA1^＊＋B1YB1^＊=C1 A2XA2^＊＋B2YB2^＊=C2，其中Ci=Ci^＊，i=1，2。通过矩阵秩的方法得到了该方程组有公共Hermitian解x，y的一种新的存在性条件，以及方程组有单独的公共Hermitian解x或y的充分必要条件。

英文摘要：

The matrix equations are {A1XA1^＊＋B1YB1^＊=C1 A2XA2^＊＋B2YB2^＊=C2, studied, whereCi=Ci^＊,i=1,2. By the rank of matrix,a necessary and sufficient condition is given for the matrix equations to have a pair of common Hermitian solutions X and Y. As a consequence, we have derived the conditions for the matrix equations to have Hermitian solution X or Y re- spectively by ranks.