Least-squares bihermitian and skew bihermitian solutions of the quaternion matrix equation AXB = C

Abstract

In this paper, by using the special structure of the real representation of quaternion matrices, the properties of the Moore–Penrose inverse of quaternion matrices and the Kronecker product of matrices, we study least-squares bihermitian and skew bihermitian solutions of the quaternion matrix equation AXB = C, respectively. First we study the special structures of quaternion bihermitian and skew bihermitian matrices. Then the problem of solving the least-squares bihermitian and skew bihermitian solutions of the quaternion matrix equation AXB = C can be transformed into particular real linear system using these special structures. We deduce the form of all least-squares bihermitian and skew bihermitian solutions of the quaternion matrix equation AXB = C and propose real structure-preserving algorithms to compute the minimal norm least-squares bihermitian and skew bihermitian solution. All the computation process only involve real arithmetic. Numerical examples are provided to verify the effectiveness of our algorithms. Particularly, we also propose the conditions and computation method of bihermitian and skew bihermitian solutions of quaternion matrix equation AXB = C.