# A new rank formula for idempotent matrices with applications

Yong Ge Tian; George P. H. Styan

Commentationes Mathematicae Universitatis Carolinae (2002)

- Volume: 43, Issue: 2, page 379-384
- ISSN: 0010-2628

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topTian, Yong Ge, and Styan, George P. H.. "A new rank formula for idempotent matrices with applications." Commentationes Mathematicae Universitatis Carolinae 43.2 (2002): 379-384. <http://eudml.org/doc/248988>.

@article{Tian2002,

abstract = {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.},

author = {Tian, Yong Ge, Styan, George P. H.},

journal = {Commentationes Mathematicae Universitatis Carolinae},

keywords = {Drazin inverse; group inverse; idempotent matrix; inner inverse; rank; tripotent matrix; Drazin inverse; group inverse; idempotent matrix; inner inverse; rank; tripotent matrix},

language = {eng},

number = {2},

pages = {379-384},

publisher = {Charles University in Prague, Faculty of Mathematics and Physics},

title = {A new rank formula for idempotent matrices with applications},

url = {http://eudml.org/doc/248988},

volume = {43},

year = {2002},

}

TY - JOUR

AU - Tian, Yong Ge

AU - Styan, George P. H.

TI - A new rank formula for idempotent matrices with applications

JO - Commentationes Mathematicae Universitatis Carolinae

PY - 2002

PB - Charles University in Prague, Faculty of Mathematics and Physics

VL - 43

IS - 2

SP - 379

EP - 384

AB - 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.

LA - eng

KW - Drazin inverse; group inverse; idempotent matrix; inner inverse; rank; tripotent matrix; Drazin inverse; group inverse; idempotent matrix; inner inverse; rank; tripotent matrix

UR - http://eudml.org/doc/248988

ER -

## References

top- 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''.
- 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]. (2000)
- Tian Y., Completing block matrices with maximal and minimal ranks, Linear Algebra Appl. 321 (2000), 327-345. (2000) MR1800003
- Tian Y., Styan, G.P.H., Some rank equalities for idempotent and involutory matrices, Linear Algebra Appl. 335 (2001), 101-117. (2001) MR1850817

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