Eigenvalue Problems

Abstract

Many practical problems in engineering and physics lead to eigenvalue problems. Typically, in all these problems, an overdetermined system of equations is given, say n + 1 equations for n unknowns ξ ₁, ...,ξ _n of the form

$$f\left( {X;\lambda } \right): \equiv \left[ {\begin{array}{*{20}{c}} {{{f}_{1}}\left( {{{\xi }_{1}}, \ldots ,{{\xi }_{n}};\lambda } \right)} \\ \vdots \\ {{{f}_{n}} + \left( {{{\xi }_{1}}, \ldots ,{{\xi }_{n}};\lambda } \right)} \\ \end{array} } \right] = 0$$

(6.0.1)

in which the functions f _i also depend on an additional parameter λ. Usually, (6.0.1) has a solution x = [ξ₁,...,ξ _n ]^T only for specific values λ = λ _i , i = 1, 2,..., of this parameter. These values λ _i are called eigenvalues of the eigenvalue problem (6.0.1) and a corresponding solution x = x(λ _i ) of (6.0.1), eigensolution belonging to the eigenvalue λ _i .