Least squares η-bi-Hermitian problems of the quaternion matrix equation (AXB,CXD) = (E,F)

Abstract

For any and denote . Let , where is the th column of the identity matrix of order If and is called an -bi-Hermitian matrix. If and is called an -anti-bi-Hermitian matrix. -bi-Hermitian matrix set and -anti-bi-Hermitian matrix set are denoted by and , respectively. In this paper, the least squares problems of quaternion matrix equation over and are considered. The expressions of the least squares -bi-Hermitian solution with the least norm and the least squares -anti-bi-Hermitian solution with the least norm of the quaternion matrix equation are derived, respectively.

Keywords:

• matrix equation
• least squares solution
• Moore–Penrose generalized inverse
• Kronecker product
• quaternion matrices
• -bi-Hermitian matrices

AMS Subject Classifications:

• 65F05
• 65H10
• 15A33

Acknowledgements

The authors are very much indebted to the editors and the anonymous referees for their constructive and valuable comments and suggestions which greatly improved the original manuscript of this paper.

Additional information

Funding

This work is supported by the Natural Science Foundation of China (11271117, 11301397), Guangdong Natural Science Fund of China (No.10452902001005845), and Science and Technology Project of Jiangmen City, China.