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Why generalized eigenvalues?

In the entry , applications of eigenvalues and eigenvectors we find numerous scientific fields in which eigenvalues are used to obtain solutions. Generalized_eigenvalue_problem are less widespread but are a key in the study of Vibrations. They are useful when we use the Galerkin method or Rayleigh-Ritz method to find approximate solutions of partial differential equations modeling vibrations of structures such as strings and plates; the paper of Courant (1943) is fundamental. The Finite element method is a widespread particular case.

In classical mechanics, we may find generalzed eigenvalues when we look for vibrations of multiple degrees of freedom systems close to equilibrium; the kinetic energy provides the mass matrix $M$, the potential strain energy provides the rigidity matrix $K$. To get details, for example see the first section of this article (1941, in French)

With both methods, we obtain a system of differential equations or Matrix differential equation $M{\ddot {x}}+B{\dot {x}}+Kx=0$ with the mass matrix $M$ , the damping matrix $B$ and the rigidity matrix $K$. If we neglect the damping effect, we use $B=0$, we can look for a solution of the following form $x=e^{i\omega t}u$; we obtain that $u$ and $\omega ^{2}$ are solution of the generalized eigenvalue problem $-\omega ^{2}Mu+Ku=0$

Bibliography

Weinstein, A.. 2.1 (1941): 36-55. <http:>.

liens utiles

   https://en.wikipedia.org/wiki/Finite_element_method_in_structural_mechanics
   https://en.wikipedia.org/wiki/Method_of_mean_weighted_residuals
   https://en.wikipedia.org/wiki/Galerkin_method#Matrix_form_of_Galerkin's_equation
   https://encyclopediaofmath.org/wiki/Ritz_method
   https://fr.wikipedia.org/wiki/M%C3%A9thode_de_Galerkine
   https://fr.wikipedia.org/wiki/Valeur_propre,_vecteur_propre_et_espace_propre#Th%C3%A9orie_spectrale
   https://fr.wikipedia.org/wiki/Mode_normal
   https://en.wikipedia.org/wiki/Normal_mode#Standing_waves
   https://de.wikipedia.org/wiki/Eigenmode#Normalkoordinaten
   https://fr.wikipedia.org/wiki/Probl%C3%A8me_aux_valeurs_propres_g%C3%A9n%C3%A9ralis%C3%A9
   https://en.wikipedia.org/wiki/Eigenfunction#Schr%C3%B6dinger_equation
   https://en.wikipedia.org/wiki/Rayleigh_quotient#Special_case_of_covariance_matrices
   https://en.wikipedia.org/wiki/Min-max_theorem
   https://en.wikipedia.org/wiki/Rayleigh%27s_quotient_in_vibrations_analysis
   https://en.wikipedia.org/wiki/Eigendecomposition_of_a_matrix#Generalized_eigenvalue_problem
   https://en.wikipedia.org/wiki/Semi-differentiability#Higher-dimensional_case