Solution of the eigenvalue problem for a regular pencil $\lambda A_0-A_1$ with singular matrices
Zapiski Nauchnykh Seminarov POMI, Computational methods and algorithms, Tome 70 (1977), pp. 103-123
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One considers the generalized eigenvalue problem
\begin{equation}
(A_0\lambda-A_1)x=0,
\end{equation}
when one or both matrices $A_0$, $A_1$ are singular and ker $\operatorname{ker}A_0\cap\operatorname{ker}A_1=\varnothing$ is the empty set. With the aid of the normalized process, the solving of problem (1) reduces to the solving of the eigenvalue problem of a constant matrix of order $r=\min(r_0,r_1)$, where $r_0$, $r_1$ are the ranks of the matrices $A_0$, $A_1$, which are determined at the normalized decomposition of the matrices. One gives an Algol program which performs the presented algorithm and testing examples.
@article{ZNSL_1977_70_a6,
author = {V. N. Kublanovskaya and T. Ya. Kon'kova},
title = {Solution of the eigenvalue problem for a regular pencil $\lambda A_0-A_1$ with singular matrices},
journal = {Zapiski Nauchnykh Seminarov POMI},
pages = {103--123},
publisher = {mathdoc},
volume = {70},
year = {1977},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/ZNSL_1977_70_a6/}
}
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%0 Journal Article %A V. N. Kublanovskaya %A T. Ya. Kon'kova %T Solution of the eigenvalue problem for a regular pencil $\lambda A_0-A_1$ with singular matrices %J Zapiski Nauchnykh Seminarov POMI %D 1977 %P 103-123 %V 70 %I mathdoc %U http://geodesic.mathdoc.fr/item/ZNSL_1977_70_a6/ %G ru %F ZNSL_1977_70_a6
V. N. Kublanovskaya; T. Ya. Kon'kova. Solution of the eigenvalue problem for a regular pencil $\lambda A_0-A_1$ with singular matrices. Zapiski Nauchnykh Seminarov POMI, Computational methods and algorithms, Tome 70 (1977), pp. 103-123. http://geodesic.mathdoc.fr/item/ZNSL_1977_70_a6/