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 matrix-valued function of the number
is known as the (nonlinear) eigenvalue, the vector
In the discipline of numerical linear algebra the following definition is typically used.
be a function that maps scalars to matrices.
is called an eigenvalue, and a nonzero vector
is called a left eigevector if
denotes the Hermitian transpose.
The definition of the eigenvalue is equivalent to
denotes the determinant.
is usually required to be a holomorphic function of
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
is said to have algebraic multiplicity
is the smallest integer such that the
[1][4] Definition: The geometric multiplicity of an eigenvalue
is the dimension of the nullspace of
[1][4] The following examples are special cases of the nonlinear eigenproblem.
is called a Jordan chain if
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:[1] A tuple of vectors
is a Jordan chain if and only if the function
and the root is of multiplicity at least
, where the vector valued function
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.