The η-bihermitian solution to a system of real quaternion matrix equations

Abstract

Let be the set of all matrices over the real quaternion algebra. , , , where is the conjugate of the quaternion . We call that is -Hermitian, if , ; is -bihermitian, if . We in this paper, present the solvability conditions and the general -Hermitian solution to a system of linear real quaternion matrix equations. As an application, we give the necessary and sufficient conditions for the systemto have an -bihermitian solution. We establish an expression of the -bihermitian to the system when it is solvable. We also obtain a criterion for a quaternion matrix to be -bihermitian. Moreover, we provide an algorithm and a numerical example to illustrate the theory developed in this paper.

Keywords:

• system of quaternion matrix equations
• η-Hermitian solution
• η-bihermitian solution
• quaternion involutions
• Moore–Penrose inverse

AMS Subject Classifications:

• 15A24
• 15A09
• 11R52
• 15B57
• 65F30

Notes

This research was supported by the National Natural Science Foundation of China [grant number 11171205], the Key Project of Scientific Research Innovation Foundation of Shanghai Municipal Education Commission [grant number 13ZZ080], and the Natural Science Foundation of Shanghai [grant number 11ZR1412500].