RANK EQUALITIES FOR MOORE-PENROSE INVERSE AND DRAZIN INVERSE OVER QUATERNION

In this paper, we consider the ranks of four real matrices G(i)(i = 0, 1, 2, 3) in M-dagger, where M = M-0 + M(1)i +M(2)j +M(3)k is an arbitrary quaternion matrix, and M-dagger = G(0) + G(1)i + G(2)j + G(3)k is the Moore-Penrose inverse of M. Similarly, the ranks of four real matrices in Drazin inverse of a quaternion matrix are also presented. As applications, the necessary and sufficient conditions for M-dagger is pure real or pure imaginary Moore-Penrose inverse and N-D is pure real or pure imaginary Drazin inverse are presented, respectively.