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The least squares η-Hermitian problems of quaternion matrix equation AHXA + BHYB=C

Yuan Shi-Fang (Wuyi University, School of Mathematics and Computational Science, Jiangmen, P. R. China)
Wang Qing-Wen (Shanghai University, Department of Mathematics, Shanghai, P. R. China)
Xiong Zhi-Ping (Wuyi University, School of Mathematics and Computational Science, Jiangmen, P. R. China)

For any A=A1+A2jQnxn and η {i,j,k} denote AηH = -ηAHη. If AηH = A,A is called an η-Hermitian matrix. If AηH =-A,A is called an η-anti-Hermitian matrix. Denote η-Hermitian matrices and η-anti-Hermitian matrices by ηHQnxn and ηAQnxn, respectively. In this paper, we consider the least squares η-Hermitian problems of quaternion matrix equation AHXA+ BHYB = C by using the complex representation of quaternion matrices, the Moore-Penrose generalized inverse and the Kronecker product of matrices. We derive the expressions of the least squares solution with the least norm of quaternion matrix equation AHXA + BHYB = C over [X,Y]  ηHQnxn x ηHQkxk, [X,Y]  ηAQnxn x ηAQkxk, and [X,Y]  ηHQnxn x ηAQkxk, respectively.

Keywords: Matrix equation, Least squares solution, Moore-Penrose generalized inverse, Kronecker product, η-Hermitian matrices