The (anti)-η-Hermitian solution of quaternion linear system

Abstract

Based on semi-tensor product of quaternion matrices, the main purpose of this paper is to explore a new conclusion about vector representation operation of quaternion matrix that can be used to solve quaternion matrix equation, and we combine the structure matrix of the multiplication of quaternion to study the different matrix representation of quaternion matrix, the matrix representation that satisfies certain conditions is defined as L-representation. Based on these conclusions we study the sufficient and necessary conditions for the existence of (anti)-η-Hermitian solutions of the system of equations (1.1) over the quaternion algebra, and for some quaternion matrices with special structure, we use GH-representation method to extract their independent elements. Then the special solution of quaternion linear systemis obtained by using classical matrix theory. The effectiveness of the algorithm is verified, and the comparison with the algorithms in reference [24], [30] and [31] shows that the algorithm in this paper is more efficient. And the application in time-varying linear system of the presented algorithm is represented.