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

Abstract

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.