In this paper, we apply the semi-tensor product of matrices and the real vector representation of a quaternion matrix to find the least squares lower (upper) triangular Toeplitz solution of \(AX-XB=C\), \(AXB-CX^{T}D=E\) and (anti)centrosymmetric solution of \(AXB-CYD=E\). And the expressions of the least squares lower (upper) triangular Toeplitz and (anti)centrosymmetric solution for the studied equations are derived. Additionally, the necessary and sufficient conditions for the existence of solutions and general expression of the studied equations are given. Eventually, some numerical examples are provided for showing the validity and superiority of our method.

在本文中，我们应用矩阵的半张量积和四元数矩阵的实向量表示来找到 \（AX-XB=C\）、\（AXB-CX^{T}D=E\） 和 \（AXB-CYD=E\） 的（反）中心对称解的最小二乘下（上）三角托普利茨解。并推导出所研究方程的最小二乘法、下（上）三角 Toeplitz 和（反）中心对称解的表达式。此外，还给出了解存在的必要和充分条件以及所研究方程的一般表达式。最后，提供了一些数值示例来证明我们方法的有效性和优越性。