Abstract
Solutions to eigenvalue problems come in two parts, an eigenvalue and an eigenvector, and these solution pairs occur at discrete points in the range of possible eigenvalues. Multiparameter eigenvalue problems similarly have solutions that have a dimension smaller by 1 than the space of the eigenvalues - solutions to a 2-parameter problem are discrete curves in a plane, and in general, solutions to an n-parameter problem are hypersurfaces in an n dimensional space. These curves/surfaces/hypersurfaces are eigenvalue interaction curves (/surfaces, etc.), and they might be flat. An unchanging eigenvector leads to a flat interaction, almost trivially. This paper addresses the question if an interaction is flat (in particular, if an interaction curve is straight), what conditions does this place on the eigenvector?
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References
Aitken AC (1956) Determinants and Matrices, 9th ed, Oliver and Boyd, Edinburgh and London; Interscience Publishers, New York
Parlett BN (1980) The symmetric eigenvalue problem. Prentice Hall, Englewood Cliffs
Wilkinson JH (1965) The algebraic eigenvalue problem. Oxford UP, Oxford
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The online version of the original article can be found at http://dx.doi.org/10.1007/s00466-005-0675-4
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Lawther, R. On the straightness of eigenvalue interactions. Comput Mech 37, 362–368 (2006). https://doi.org/10.1007/s00466-005-0746-6
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DOI: https://doi.org/10.1007/s00466-005-0746-6