Rank equalities for Moore-Penrose inverse and Drazin‎ ‎inverse over quaternion

Abstract

‎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.