Yongge Tian, George P.H. Styan
A new rank formula for idempotent matrices with applications

Comment.Math.Univ.Carolinae 43,2 (2002) 379-384.

Abstract:It is shown that $$ \text {rank}(P^*AQ) = \text {rank}(P^*A) + \text {rank}(AQ) - \text {rank}(A), $$ where $A$ is idempotent, $[P,Q]$ has full row rank and $P^*Q = 0$. Some applications of the rank formula to generalized inverses of matrices are also presented.

Keywords: Drazin inverse, group inverse, idempotent matrix, inner inverse, rank, tripotent matrix
AMS Subject Classification: 15A03, 15A09