Eigenvalue Problems

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 ξ₁,..., ξ _n of the form 6.0.1 $$F(x;\lambda ): \equiv \left[ {\begin{array}{*{20}{c}} {{f_1}({\xi _1}, \ldots ,{\xi _n};\lambda )} \\ { \ldots \ldots \ldots \ldots \ldots \ldots \ldots } \\ {{f_{n + 1}}({\xi _1}, \ldots ,{\xi _n};\lambda )} \end{array}} \right] = 0,$$ 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 .