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 is a vector, and ' is a matrix-valued function of the number . The number is known as the (nonlinear) eigenvalue, the vector as the (nonlinear) eigenvector, and as the eigenpair. The matrix is singular at an eigenvalue .
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 eigenvector if . Moreover, a nonzero vector is called a left eigenvector if , where the superscript denotes the Hermitian transpose. The definition of the eigenvalue is equivalent to , where denotes the determinant.
The function ' is usually required to be a holomorphic function of (in some domain ).
In general, 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 such that . Otherwise it is said to be singular.
Definition: An eigenvalue is said to have algebraic multiplicity if is the smallest integer such that the th derivative of with respect to , in is nonzero. In formulas that but for .
Definition: The geometric multiplicity of an eigenvalue is the dimension of the nullspace of .
The following examples are special cases of the nonlinear eigenproblem.
Definition: Let be an eigenpair. A tuple of vectors is called a Jordan chain iffor , where denotes the th derivative of with respect to and evaluated in . The vectors are called generalized eigenvectors, is called the length of the Jordan chain, and the maximal length a Jordan chain starting with is called the rank of .
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 for , where the vector valued function is defined as
Eigenvector nonlinearities is a related, but different, form of nonlinearity that is sometimes studied. In this case the function maps vectors to matrices, or sometimes hermitian matrices to hermitian matrices.