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Keywords:
Drazin inverse; group inverse; idempotent matrix; inner inverse; rank; tripotent matrix
Summary:
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.
References:
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