四元数矩阵方程最小二乘Toeplitz解的半张量积方法

Semi-tensor product method for solving least square Toeplitz solutions of quaternion matrix equation

研究了四元数矩阵方程■的最小二乘Toeplitz解和Hermitian Toeplitz解的问题.联合使用四元数矩阵的实向量表示方法和矩阵的半张量积方法,将所研究的问题转化为实矩阵方程.根据Toeplitz矩阵以及Hermitian Toeplitz矩阵的结构特征,提取了矩阵中的有效元素,构造了新的解向量,降低了所研究问题的复杂度.得到了方程存在Toeplitz解和Hermitian Toeplitz解的条件,并给出Toeplitz解和Hermitian Toeplitz解的一般形式.通过数值算例说明了方法的精度和算法的可行性.

Least square Toeplitz solutions and Hermitian Toeplitz solutions of the quaternion matrix equation ■ are studied.Utilizing real vector representation of the quaternion matrix and semi-sensor product theory,the quaternion matrix equation is transformed into its equivalent real matrix equation.Considering the structural characteristics of the Toeplitz matrix and Hermitian Toeplitz matrix,independent elements of the solution matrix are extracted to reconstruct a new solution vector,thus the computational complexity of the problem is reduced.The existing conditions of Toeplitz solutions and Hermitian Toeplitz solutions of the equation are obtained,and the general solutions of the equation are given.Finally,a numerical example is given to demonstrate the precision degree and effectiveness of the algorithm.