Geodesic
A new rank formula for idempotent matrices with applications
Commentationes Mathematicae Universitatis Carolinae, Tome 43 (2002) no. 2, pp. 379-384.

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It is shown that $$ \text{\rm rank}(P^*AQ) = \text{\rm rank}(P^*A) + \text{\rm rank}(AQ) - \text{\rm rank}(A), $$ 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.
Classification : 15A03, 15A09
Keywords: Drazin inverse; group inverse; idempotent matrix; inner inverse; rank; tripotent matrix 
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Tian, Yongge; Styan, George P. H. A new rank formula for idempotent matrices with applications. Commentationes Mathematicae Universitatis Carolinae, Tome 43 (2002) no. 2, pp. 379-384. http://geodesic.mathdoc.fr/item/CMUC_2002__43_2_a14/