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

Abstract

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It is shown that rank ( P * A Q ) = rank ( P * A ) + rank ( A Q ) - 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.

How to cite

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Tian, 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

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  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]. (2000) 
  3. Tian Y., Completing block matrices with maximal and minimal ranks, Linear Algebra Appl. 321 (2000), 327-345. (2000) MR1800003
  4. 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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