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
where formula_2 is a vector, and "formula_3" is a matrix-valued function of the number formula_4. The number formula_4 is known as the (nonlinear) eigenvalue, the vector formula_6 as the (nonlinear) eigenvector, and formula_7 as the eigenpair. The matrix formula_8 is singular at an eigenvalue formula_4.
Definition.
In the discipline of numerical linear algebra the following definition is typically used. 
Let formula_10, and let formula_11 be a function that maps scalars to matrices. A scalar formula_12 is called an "eigenvalue", and a nonzero vector formula_13is called a "right eigevector" if formula_14. Moreover, a nonzero vector formula_15is called a "left eigevector" if formula_16, where the superscript formula_17 denotes the Hermitian transpose. The definition of the eigenvalue is equivalent to formula_18, where formula_19 denotes the determinant.
The function "formula_3" is usually required to be a holomorphic function of formula_4 (in some domain formula_22).
In general, formula_8 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 formula_24 such that formula_25. Otherwise it is said to be "singular".
Definition: An eigenvalue formula_4 is said to have "algebraic multiplicity" formula_27 if formula_27 is the smallest integer such that the formula_27th derivative of formula_30 with respect to formula_31, in formula_4 is nonzero. In formulas that formula_33 but formula_34 for formula_35.
Definition: The "geometric multiplicity" of an eigenvalue formula_4 is the dimension of the nullspace of formula_8.
Special cases.
The nonlinear eigenproblem is a generalization of the The following examples are special cases of the nonlinear eigenproblem.
Jordan chains.
Definition: Let formula_46 be an eigenpair. A tuple of vectors formula_47 is called a "Jordan chain" ifformula_48for formula_49, where formula_50 denotes the formula_27th derivative of formula_3 with respect to formula_4 and evaluated in formula_54. The vectors formula_55 are called "generalized eigenvectors", formula_56 is called the "length" of the Jordan chain, and the maximal length a Jordan chain starting with formula_57 is called the "rank" of formula_57.
Theorem: A tuple of vectors formula_47 is a Jordan chain if and only if the function formula_60 has a root in formula_54 and the root is of multiplicity at least formula_62 for formula_63, where the vector valued function formula_64 is defined asformula_65
Eigenvector nonlinearity.
Eigenvector nonlinearities is a related, but different, form of nonlinearity that is sometimes studied. In this case the function formula_3 maps vectors to matrices, or sometimes hermitian matrices to hermitian matrices. 