TITLE:
Cyclic Solution and Optimal Approximation of the Quaternion Stein Equation
AUTHORS:
Guangmei Liu, Yanting Zhang, Yiwen Yao, Jingpin Huang
KEYWORDS:
Quaternion Field, Stein Equation, Cyclic Matrix, Complex Decomposition, Real Decomposition, Optimal Approximation
JOURNAL NAME:
Journal of Applied Mathematics and Physics,
Vol.11 No.11,
November
30,
2023
ABSTRACT: In this paper, two different methods are used to study the cyclic structure solution and the optimal approximation of the quaternion Stein equation AXB - X = F . Firstly, the matrix equation equivalent to the target structure matrix is constructed by using the complex decomposition of the quaternion matrix, to obtain the necessary and sufficient conditions for the existence of the cyclic solution of the equation and the expression of the general solution. Secondly, the Stein equation is converted into the Sylvester equation by adding the necessary parameters, and the condition for the existence of a cyclic solution and the expression of the equation’s solution are then obtained by using the real decomposition of the quaternion matrix and the Kronecker product of the matrix. At the same time, under the condition that the solution set is non-empty, the optimal approximation solution to the given quaternion circulant matrix is obtained by using the property of Frobenius norm property. Numerical examples are given to verify the correctness of the theoretical results and the feasibility of the proposed method.