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 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.

Definition

In the discipline of numerical linear algebra the following definition is typically used. Let, and let be a function that maps scalars to matrices. A scalar is called an eigenvalue, and a nonzero vector is called a right eigevector if. Moreover, a nonzero vector is called a left eigevector if, where the superscript ^H denotes the Hermitian transpose. The definition of the eigenvalue is equivalent to, 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. 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 kth derivative of \det(M (z)) with respect to z, in \lambda is nonzero. In formulas that but for. Definition: The geometric multiplicity of an eigenvalue \lambda is the dimension of the nullspace of M (\lambda).

Special cases

The following examples are special cases of the nonlinear eigenproblem.

Jordan chains

Definition: Let be an eigenpair. A tuple of vectors is called a Jordan chain iffor, where denotes the kth derivative of M with respect to \lambda and evaluated in. The vectors 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 is a Jordan chain if and only if the function has a root in and the root is of multiplicity at least \ell for, where the vector valued function is defined as

Mathematical software

Eigenvector nonlinearity

Eigenvector nonlinearities is a related, but different, form of nonlinearity that is sometimes studied. In this case the function M maps vectors to matrices, or sometimes hermitian matrices to hermitian matrices.