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.
@article{119327,
author = {Yong Ge Tian and George P. H. Styan},
title = {A new rank formula for idempotent matrices with applications},
journal = {Commentationes Mathematicae Universitatis Carolinae},
volume = {43},
year = {2002},
pages = {379-384},
zbl = {1090.15001},
mrnumber = {1922135},
language = {en},
url = {http://dml.mathdoc.fr/item/119327}
}
Tian, Yong Ge; Styan, George P. H. A new rank formula for idempotent matrices with applications. Commentationes Mathematicae Universitatis Carolinae, Tome 43 (2002) pp. 379-384. http://gdmltest.u-ga.fr/item/119327/
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