一类混合型交换四元数矩阵实表示的性质及应用

在引入混合型交换四元数及混合型交换四元数矩阵概念的基础上，首先，证明了混合型交换四元数和实数域上的4阶矩阵是同构的，将对混合型交换四元数的研究转化为对实数域上4阶矩阵的研究.其次，在混合型交换四元数矩阵和实数域上4n阶矩阵同构的基础上，将对混合型交换四元数矩阵的研究转化为对实数域上4n阶矩阵的研究.利用实矩阵的性质得到混合型交换四元数矩阵实表示的系列性质，并给出了混合型交换四元数矩阵可逆的等价条件.以混合型交换四元数矩阵实表示的性质为基础，得到混合型交换四元数矩阵复特征值的个数及特征值存在的充分必要条件，并将实数域上的盖尔圆盘定理推广到混合型交换四元数矩阵上.最后，利用具体的数值算例验证了混合型交换四元数矩阵盖尔圆盘定理的正确性和有效性.

On Property and Application of Real Representation of a Mixed Type Commutative Quaternion Matrix

In this paper, studies have been done on the basis of introducing the concept of mixed type commutative quaternion and mixed type commutative quaternion matrix. Firstly, it is proved that the mixed type commutative quaternion and the fourth-order matrix on the real field are isomorphic. The study of mixed type commutative quaternion is transformed into the study of the fourth-order matrix on the real field. Secondly, on the basis of the isomorphism of the 4n th-order matrix in the real field and the mixed type commutative quaternion matrix, the study of mixed type commutative quaternion matrix is transformed into the study of the 4n th-order matrix on the real field. By means of the properties of real matrices to obtain the series properties of the real representation of mixed type commutative quaternion matrices, and the equivalent condition for the reversibility of the mixed type commutative quaternion matrix is given. Based on the property of the real representation of mixed type commutative quaternion matrix, the number of complex eigenvalues of mixed type commutative quaternion matrix and the sufficient and necessary conditions for the existence of eigenvalues are obtained. The Gerschgorin disk theorem on the real field is extended to the mixed type commutative quaternion matrix. Finally, the correctness and validity of the Gerschgorin disk theorem for mixed type commutative quaternion matrix is verified by a numerical example.