Nonlinear eigenproblem

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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 .

Definition[edit]

In the discipline of numerical linear algebra the following definition is typically used.[1][2][3][4]

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 denotes the Hermitian transpose. The definition of the eigenvalue is equivalent to , where denotes the determinant.[1]

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.[1][4]

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 .[1][4]

Definition: The geometric multiplicity of an eigenvalue is the dimension of the nullspace of .[1][4]

Special cases[edit]

The following examples are special cases of the nonlinear eigenproblem.

  • The (ordinary) eigenvalue problem:
  • The generalized eigenvalue problem:
  • The quadratic eigenvalue problem:
  • The polynomial eigenvalue problem:
  • The rational eigenvalue problem: where are rational functions.
  • The delay eigenvalue problem: where are given scalars, known as delays.

Jordan chains[edit]

Definition: Let be an eigenpair. A tuple of vectors is called a Jordan chain if

for , 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 .[1][4]


Theorem:[1] 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 nonlinearity[edit]

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.[5][6]

References[edit]

  1. ^ a b c d e f g Güttel, Stefan; Tisseur, Françoise (2017). "The nonlinear eigenvalue problem" (PDF). Acta Numerica. 26: 1–94. doi:10.1017/S0962492917000034. ISSN 0962-4929. S2CID 46749298.
  2. ^ Ruhe, Axel (1973). "Algorithms for the Nonlinear Eigenvalue Problem". SIAM Journal on Numerical Analysis. 10 (4): 674–689. doi:10.1137/0710059. ISSN 0036-1429. JSTOR 2156278.
  3. ^ Mehrmann, Volker; Voss, Heinrich (2004). "Nonlinear eigenvalue problems: a challenge for modern eigenvalue methods". GAMM-Mitteilungen. 27 (2): 121–152. doi:10.1002/gamm.201490007. ISSN 1522-2608.
  4. ^ a b c d e Voss, Heinrich (2014). "Nonlinear eigenvalue problems" (PDF). In Hogben, Leslie (ed.). Handbook of Linear Algebra (2 ed.). Boca Raton, FL: Chapman and Hall/CRC. ISBN 9781466507289.
  5. ^ Jarlebring, Elias; Kvaal, Simen; Michiels, Wim (2014-01-01). "An Inverse Iteration Method for Eigenvalue Problems with Eigenvector Nonlinearities". SIAM Journal on Scientific Computing. 36 (4): A1978–A2001. arXiv:1212.0417. doi:10.1137/130910014. ISSN 1064-8275. S2CID 16959079.
  6. ^ Upadhyaya, Parikshit; Jarlebring, Elias; Rubensson, Emanuel H. (2021). "A density matrix approach to the convergence of the self-consistent field iteration". Numerical Algebra, Control & Optimization. 11 (1): 99. doi:10.3934/naco.2020018. ISSN 2155-3297.

Further reading[edit]