Nonlinear eigenproblem

In mathematics, a nonlinear eigenproblem, sometimes nonlinear eigenvalue problem, is a generalization of the (ordinary) eigenvalue problem to equations that depend nonlinearly on the eigenvalue. Specifically, it refers to equations of the form M (\lambda) x = 0 , where x\neq0 is a vector, and M is a matrix-valued function of the number \lambda . The number \lambda is known as the (nonlinear) eigenvalue, the vector x as the (nonlinear) eigenvector, and (\lambda,x) as the eigenpair. The matrix M (\lambda) is singular at an eigenvalue \lambda .

In mathematics, a nonlinear eigenproblem, sometimes nonlinear eigenvalue problem, is a generalization of the (ordinary) eigenvalue problem to equations that depend nonlinearly on the eigenvalue. Specifically, it refers to equations of the form M (\lambda) x = 0 , where x\neq0 is a vector, and M is a matrix-valued function of the number \lambda . The number \lambda is known as the (nonlinear) eigenvalue, the vector x as the (nonlinear) eigenvector, and (\lambda,x) as the eigenpair. The matrix M (\lambda) is singular at an eigenvalue \lambda . Let \Omega \subseteq \Complex , and let M : \Omega \rightarrow \Complex^{n\times n} be a function that maps scalars to matrices. A scalar \lambda \in \Complex is called an eigenvalue, and a nonzero vector x \in \Complex^n is called a right eigenvector if M (\lambda) x = 0 . Moreover, a nonzero vector y \in \Complex^n is called a left eigenvector if y^H M (\lambda) = 0^H , where the superscript ^H denotes the Hermitian transpose. The definition of the eigenvalue is equivalent to \det(M (\lambda)) = 0 , where \det() denotes the determinant. The function M is usually required to be a holomorphic function of \lambda (in some domain \Omega ). In general, M (\lambda) could be a linear map, but most commonly it is a finite-dimensional, usually square, matrix. Definition: The problem is said to be regular if there exists a z\in\Omega such that \det(M (z)) \neq 0 . Otherwise it is said to be singular. Definition: An eigenvalue \lambda is said to have algebraic multiplicity k if k is the smallest integer such that the k th derivative of \det(M (z)) with respect to z , in \lambda is nonzero. In formulas that \left.\frac{d^k \det(M (z))}{d z^k} \right|_{z=\lambda} \neq 0 but \left.\frac{d^\ell \det(M (z))}{d z^\ell} \right|_{z=\lambda} = 0 for \ell=0,1,2,\dots, k-1 . Definition: The geometric multiplicity of an eigenvalue \lambda is the dimension of the nullspace of M (\lambda) . The (ordinary) eigenvalue problem: M (\lambda) = A-\lambda I. The generalized eigenvalue problem: M (\lambda) = A-\lambda B. The quadratic eigenvalue problem: M (\lambda) = A_0 + \lambda A_1 + \lambda^2 A_2. The polynomial eigenvalue problem: M (\lambda) = \sum_{i=0}^m \lambda^i A_i. The rational eigenvalue problem: M (\lambda) = \sum_{i=0}^{m_1} A_i \lambda^i + \sum_{i=1}^{m_2} B_i r_i(\lambda), where r_i(\lambda) are rational functions. The delay eigenvalue problem: M (\lambda) = -I\lambda + A_0 +\sum_{i=1}^m A_i e^{-\tau_i \lambda}, where \tau_1,\tau_2,\dots,\tau_m are given scalars, known as delays. Definition: Let (\lambda_0,x_0) be an eigenpair. A tuple of vectors (x_0,x_1,\dots, x_{r-1})\in\Complex^n\times\Complex^n\times\dots\times\Complex^n is called a Jordan chain if \sum_{k=0}^{\ell} M^{(k)} (\lambda_0) x_{\ell - k} = 0 , for \ell = 0,1,\dots , r-1 , where M^{(k)}(\lambda_0) denotes the k th derivative of M with respect to \lambda and evaluated in \lambda=\lambda_0 . The vectors x_0,x_1,\dots, x_{r-1} are called generalized eigenvectors, r is called the length of the Jordan chain, and the maximal length a Jordan chain starting with x_0 is called the rank of x_0 . Theorem: A tuple of vectors (x_0,x_1,\dots, x_{r-1})\in\Complex^n\times\Complex^n\times\dots\times\Complex^n is a Jordan chain if and only if the function M(\lambda) \chi_\ell (\lambda) has a root in \lambda=\lambda_0 and the root is of multiplicity at least \ell for \ell=0,1,\dots,r-1 , where the vector valued function \chi_\ell (\lambda) is defined as \chi_\ell(\lambda) = \sum_{k=0}^\ell x_k (\lambda-\lambda_0)^k.

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