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EIGENVALUES

DOI: 10.1615/AtoZ.e.eigenvalues

Eigenvalue of the operator L is the value of the parameter λ (complex or real) for which the equation Lu = λu has a nonzero solution. The appropriate nonzero solutions are called eigenfunctions of the operator L, corresponding to eigenvalue λ. L nonlinear operators are usually considered as differential, integral, etc. A set of eigenvalues is a discrete spectrum of the operator L. Eigenfunctions belonging to different eigenvalues are linearly independent.

The Hermitian (conjugate) linear operators [for instance, the differential operator involved in the stationary equations of heat transfer and diffusion L = div(k · gradT) play an important part in solving problems of heat transfer. If the operator L is self-conjugate, then all its eigenvalues are real. Eigenfunctions corresponding to different eigenvalues are mutually orthogonal. If a self-conjugate operator L has a purely discrete spectrum, then it has a complete orthonormal sequence of eigenvalues.

The expansion of functions into a series in terms of the orthonormal sequence of eigenfunctions (the Fourier Series) is of paramount importance in solving problems of hydrodynamics and heat transfer (in analyzing computational algorithms, in particular).

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