Zapiski Nauchnykh Seminarov POMI, Computational methods and algorithms. Part XXI, Tome 359 (2008), pp. 31-35
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Kh. D. Ikramov. Gaussian elimination and the ranks of the components in the Cartesian decomposition of a matrix. Zapiski Nauchnykh Seminarov POMI, Computational methods and algorithms. Part XXI, Tome 359 (2008), pp. 31-35. http://geodesic.mathdoc.fr/item/ZNSL_2008_359_a2/
@article{ZNSL_2008_359_a2,
author = {Kh. D. Ikramov},
title = {Gaussian elimination and the ranks of the components in the {Cartesian} decomposition of a~matrix},
journal = {Zapiski Nauchnykh Seminarov POMI},
pages = {31--35},
year = {2008},
volume = {359},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/ZNSL_2008_359_a2/}
}
TY - JOUR
AU - Kh. D. Ikramov
TI - Gaussian elimination and the ranks of the components in the Cartesian decomposition of a matrix
JO - Zapiski Nauchnykh Seminarov POMI
PY - 2008
SP - 31
EP - 35
VL - 359
UR - http://geodesic.mathdoc.fr/item/ZNSL_2008_359_a2/
LA - ru
ID - ZNSL_2008_359_a2
ER -
%0 Journal Article
%A Kh. D. Ikramov
%T Gaussian elimination and the ranks of the components in the Cartesian decomposition of a matrix
%J Zapiski Nauchnykh Seminarov POMI
%D 2008
%P 31-35
%V 359
%U http://geodesic.mathdoc.fr/item/ZNSL_2008_359_a2/
%G ru
%F ZNSL_2008_359_a2
Let $A=B+iC$, where $B=B^*$, $C=C^*$, be the Cartesian decomposition of an $n\times n$ matrix $A$, and let the component $B$ (or $C$) have rank $r. It is shown that for a nonsingular $A$, the inverse $A^{-1}$ has an analogous property. This implies that all the (correctly defined) Schur complements in $A$ have Cartesian decompositions with component $B$ (or $C$) of rank $\le r$. The active submatrix at each step of the Gaussian elimination applied to $A$ is the Schur complement of the appropriate leading principal submatrix. Bibl. – 2 titles.
