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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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