The spectrum of a modified linear pencil
Date
2003
Authors
Elhay, S.
Golub, G.
Ram, Y.
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Journal article
Citation
Computers and Mathematics with Applications, 2003; 46(8-9):1413-1426
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Abstract
Suppose the spectrum of a symmetric definite linear pencil is known, This paper addresses the question of what can be said about the spectrum when scalar multiples of a rank-one update are added to each matrix in the pencil. The secular equation for this problem is derived, and from it, a certain separation property is found which gives insight into the connection between the eigenvalues before and after modification. In the context of structural dynamics, the result characterises the behaviour of a finite-dimensional vibrating system undergoing mass and stiffness modifications. The result also leads to applications such as a divide and conquer algorithm for the eigenvalues of the modified system (so-called matrix tearing) and spectral shifting. An illustrative example is also given. © 2003 Elsevier Ltd. All rights reserved.