Journal Article10.13001/1081-3810.1322
On the singular two-parameter eigenvalue problem ∗
Andrej Muhič,Bor Plestenjak +1 more
TL;DR: It is shown that the simple finite regular eigenvalues of a two-parameter eigen value problem and the associated system of generalized eigenvalue problems agree.
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Abstract: In the 1960s, Atkinson introduced an abstract algebraic setting for multiparameter eigenvalue problems. He showed that a nonsingular multiparameter eigenvalue problem is equivalent to the associated system of generalized eigenvalue problems. Many theoretical results and numerical methods for nonsingular multiparameter eigenvalue problems are based on this relation. In this paper, the above relation to singular two-parameter eigenvalue problems is extended, and it is shown that the simple finite regular eigenvalues of a two-parameter eigenvalue problem and the associated system of generalized eigenvalue problems agree. This enables one to solve a singular two-parameter eigenvalue problem by computing the common regular eigenvalues of the associated system of two singular generalized eigenvalue problems.
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Citations
On linearizations of the quadratic two-parameter eigenvalue problems
TL;DR: In this paper, the quadratic two-parameter eigenvalue problem (QMEP) is transformed into a nonsingular five-parameters eigen value problem.
32
Polynomial two-parameter eigenvalue problems and matrix pencil methods for stability of delay-differential equations
TL;DR: In this article, a different derivation of various matrix pencil methods is presented using a unifying framework of a new type of eigenvalue problem: the polynomial two-parameter eigen value problem.
31
Solving singular generalized eigenvalue problems by a rank-completing perturbation
TL;DR: In this paper, a simple method to compute the eigenvalues of singular pencils, based on one perturbation of the original problem of a certain specific rank, is proposed.
23
On linearizations of the quadratic two-parameter eigenvalue problems
Michiel E. Hochstenbach,Andrej Muhič,Bor Plestenjak +2 more
- 01 Jan 2011
TL;DR: In this article, the quadratic two-parameter eigenvalue problem (QMEP) is solved by formulating an associated linear multiparameter Eigenvalue (MLE) problem.
22
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