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