Study of diagonalization on skew self-conjugate matrix in quaternion field

This paper made a further research based on the extended unitary diagonalization of skew self-conjugate matrix. Based on the definition of skew selfconjugate matrix, we discussed some properties of skew self-conjugate matrix, and gave the necessary and sufficient condition for determining whether ; is a right eigenvalue of matrix in the quaternion field. By means of the Schur theorem in real quaternion field, we proved that skew self-conjugate can be extended unitary diagonalized. Furthermore, if A and B are invertible and commutative, then A + B and A^-1 + B^-1 can be extended unitary diagonalized at the same time.

1. Introduction

Quaternion matrices play important roles in quantum physics^[1], Diagonalization theories are important parts in theories of quaternion matrices. Paper [2] studied a new proof of extended unitary diagonalizable of self-conjugate matrix in quaternion field. In paper [3], necessary and sufficient conditions of generalized diagonalization of quaternion matrices were studied and a method of generalized diagonalization was obtained. Paper [4] gave necessary and sufficient conditions for determining whether a quaternion matrix is a diagonalization matrix and proved Schur theorem over quaternion field. In this paper, we study some properties and the extended unitary diagonalization of skew self-conjugate matrix.