Tian, Yongge; Styan, George P. H. A new rank formula for idempotent matrices with applications. (English) Zbl 1090.15001 Commentat. Math. Univ. Carol. 43, No. 2, 379-384 (2002). 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. Cited in 6 Documents MSC: 15A03 Vector spaces, linear dependence, rank, lineability 15A09 Theory of matrix inversion and generalized inverses Keywords:Drazin inverse; group inverse; idempotent matrix; inner inverse; rank; tripotent matrix PDFBibTeX XMLCite \textit{Y. Tian} and \textit{G. P. H. Styan}, Commentat. Math. Univ. Carol. 43, No. 2, 379--384 (2002; Zbl 1090.15001) Full Text: EuDML EMIS