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A new rank formula for idempotent matrices with applications. (English) Zbl 1090.15001
Summary: 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.

##### MSC:
 15A03 Vector spaces, linear dependence, rank, lineability 15A09 Theory of matrix inversion and generalized inverses
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