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Drazin inverse; group inverse; idempotent matrix; inner inverse; rank; tripotent matrix
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.
[1] Drury S.W., Liu S., Lu C.Y., Puntanen S., Styan G.P.H.: Some comments on several matrix inequalities with applications to canonical correlations: historical background and recent developments. Report A332 (December 2000), Dept. of Mathematics, Statistics and Philosophy, University of Tampere, Tampere, Finland, 63 pp. To be published in the special issue of {Sankhyā: The Indian Journal of Statistics, Series A} associated with ``An International Conference in Honor of Professor C.R. Rao on the Occasion of his 80th Birthday, Statistics: Reflections on the Past and Visions for the Future, The University of Texas at San Antonio, March 2000''.
[2] Tian Y.: Two rank equalities associated with blocks of orthogonal projector. Problem $25$-$4$. Image, The Bulletin of the International Linear Algebra Society 25 (2000), p.16 [Solutions by J.K. Baksalary & O.M. Baksalary, by H.J. Werner, and by S. Puntanen, G.P.H. Styan & Y. Tian, Image, The Bulletin of the International Linear Algebra Society 26 (2001), 6-9].
[3] Tian Y.: Completing block matrices with maximal and minimal ranks. Linear Algebra Appl. 321 (2000), 327-345. MR 1800003
[4] Tian Y., Styan, G.P.H.: Some rank equalities for idempotent and involutory matrices. Linear Algebra Appl. 335 (2001), 101-117. MR 1850817
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