Quadratic eigenvalue problem: Difference between revisions

Revision as of 11:35, 20 September 2023

Revision as of 11:35, 20 September 2023

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==Spectral theory==

A QEP is said to be <em>regular</em> if <math>\text{det} (Q(\lambda)) \not \equiv 0</math> identically. The coefficient of the <math>\lambda^{2n}</math> term in <math>\text{Detdet}(Q(\lambda))</math> is <math>\text{Detdet}(M)</math>, implying that the QEP is regular if <math>M</math> is nonsingular.

Eigenvalues at infinity and eigenvalues at 0 may be exchanged by considering the reversed polynomial, <math> \lambda^2 Q(\lambda^{-1}) = \lambda^2 K + \lambda C + M </math>. As there are <math> 2n</math> eigenvectors in a <math>n</math> dimensional space, the eigenvectors cannot be orthogonal. It is possible to have the same eigenvector attached to different eigenvalues.