| 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:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
43 |
| Issue:
|
2 |
| Year:
|
2002 |
| Pages:
|
379-384 |
| . |
| 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:
|
2009-01-08T19:22:59Z |
| Last updated:
|
2012-04-30 |
| 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), 6-9]. |
| Reference:
|
[3] Tian Y.: Completing block matrices with maximal and minimal ranks.Linear Algebra Appl. 321 (2000), 327-345. MR 1800003 |
| Reference:
|
[4] Tian Y., Styan, G.P.H.: Some rank equalities for idempotent and involutory matrices.Linear Algebra Appl. 335 (2001), 101-117. MR 1850817 |
| . |
