Least squares solution of the quaternion matrix equation with the least norm

Abstract

For any , is called the j-conjugate matrix of A. If , A is called a j-self-conjugate matrix. If , A is called an anti j-self-conjugate matrix. 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, the least squares j-self-conjugate solution with the least norm, and the least squares anti j-self-conjugate solution with the least norm of the matrix equation over the skew field of quaternions, respectively.

Keywords:

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

AMS Subject Classifications:

• 65F05
• 65H10
• 15A33

Acknowledgements

The authors would like to thank the editor-in-chief, Prof. Fuzhen Zhang and anonymous referees for their useful comments and suggestions. This work is supported by Natural Science Fund of China (No.61070150), Hunan Provincial Natural Science Fund of China (No. 09JJ6012) and Guangdong Natural Science Fund of China (No.10452902001005845).