Title:

A new rank formula for idempotent matrices with applications (English) 
Author:

Tian, Yongge 
Author:

Styan, George P. H. 
Language:

English 
Journal:

Commentationes Mathematicae Universitatis Carolinae 
ISSN:

00102628 (print) 
ISSN:

12137243 (online) 
Volume:

43 
Issue:

2 
Year:

2002 
Pages:

379384 
. 
Category:

math 
. 
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. (English) 
Keyword:

Drazin inverse 
Keyword:

group inverse 
Keyword:

idempotent matrix 
Keyword:

inner inverse 
Keyword:

rank 
Keyword:

tripotent matrix 
MSC:

15A03 
MSC:

15A09 
idZBL:

Zbl 1090.15001 
idMR:

MR1922135 
. 
Date available:

20090108T19:22:59Z 
Last updated:

20120430 
Stable URL:

http://hdl.handle.net/10338.dmlcz/119327 
. 
Reference:

[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''. 
Reference:

[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), 69]. 
Reference:

[3] Tian Y.: Completing block matrices with maximal and minimal ranks.Linear Algebra Appl. 321 (2000), 327345. MR 1800003 
Reference:

[4] Tian Y., Styan, G.P.H.: Some rank equalities for idempotent and involutory matrices.Linear Algebra Appl. 335 (2001), 101117. MR 1850817 
. 