Geodesic
A new rank formula for idempotent matrices with applications
Commentationes Mathematicae Universitatis Carolinae, Tome 43 (2002) no. 2, pp. 379-384 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

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.
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.
Classification : 15A03, 15A09
Keywords: Drazin inverse; group inverse; idempotent matrix; inner inverse; rank; tripotent matrix 
@article{CMUC_2002_43_2_a14,
     author = {Tian, Yongge and Styan, George P. H.},
     title = {A new rank formula for idempotent matrices with applications},
     journal = {Commentationes Mathematicae Universitatis Carolinae},
     pages = {379--384},
     year = {2002},
     volume = {43},
     number = {2},
     mrnumber = {1922135},
     zbl = {1090.15001},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/CMUC_2002_43_2_a14/}
}
TY  - JOUR
AU  - Tian, Yongge
AU  - Styan, George P. H.
TI  - A new rank formula for idempotent matrices with applications
JO  - Commentationes Mathematicae Universitatis Carolinae
PY  - 2002
SP  - 379
EP  - 384
VL  - 43
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/CMUC_2002_43_2_a14/
LA  - en
ID  - CMUC_2002_43_2_a14
ER  - 
%0 Journal Article
%A Tian, Yongge
%A Styan, George P. H.
%T A new rank formula for idempotent matrices with applications
%J Commentationes Mathematicae Universitatis Carolinae
%D 2002
%P 379-384
%V 43
%N 2
%U http://geodesic.mathdoc.fr/item/CMUC_2002_43_2_a14/
%G en
%F CMUC_2002_43_2_a14
Tian, Yongge; Styan, George P. H. A new rank formula for idempotent matrices with applications. Commentationes Mathematicae Universitatis Carolinae, Tome 43 (2002) no. 2, pp. 379-384. http://geodesic.mathdoc.fr/item/CMUC_2002_43_2_a14/