Nonlinear eigenproblem

A nonlinear eigenproblem is a generalization of an ordinary eigenproblem to equations that depend nonlinearly on the eigenvalue. Specifically, it refers to equations of the form:

$A(\lambda )\mathbf {x} =0,\,$

where x is a vector (the nonlinear "eigenvector") and A is a matrix-valued function of the number $\lambda$ (the nonlinear "eigenvalue"). (More generally, $A(\lambda )$ could be a linear map, but most commonly it is a finite-dimensional, usually square, matrix.)

For example, an ordinary linear eigenproblem $B\mathbf {v} =\lambda \mathbf {v}$, where B is a square matrix, corresponds to $A(\lambda )=B-\lambda I$, where I is the identity matrix.

One common case is where A is a polynomial matrix, which is called a polynomial eigenvalue problem. In particular, the specific case where the polynomial has degree two is called a quadratic eigenvalue problem or a quadratic matrix equation, and can be written in the form:

$A(\lambda )\mathbf {x} =(A_{2}\lambda ^{2}+A_{1}\lambda +A_{0})\mathbf {x} =0,\,$

in terms of the constant square matrices A_0,1,2. This can be converted into an ordinary linear generalized eigenproblem of twice the size by defining a new vector $\mathbf {y} =\lambda \mathbf {x}$. In terms of x and y, the quadratic eigenvalue problem becomes:

${\begin{pmatrix}-A_{0}&0\\0&I\end{pmatrix}}{\begin{pmatrix}\mathbf {x} \\\mathbf {y} \end{pmatrix}}=\lambda {\begin{pmatrix}A_{1}&A_{2}\\I&0\end{pmatrix}}{\begin{pmatrix}\mathbf {x} \\\mathbf {y} \end{pmatrix}},$

where I is the identity matrix. More generally, if A is a matrix polynomial of degree d then one can convert the nonlinear eigenproblem into a linear (generalized) eigenproblem of d times the size.

Besides converting them to ordinary eigenproblems, which only works if A is polynomial, there are other methods of solving nonlinear eigenproblems based on the Jacobi-Davidson algorithm or based on Newton's method (related to inverse iteration).

References

• Philippe Guillaume, "Nonlinear eigenproblems," SIAM J. Matrix. Anal. Appl. '20 (3), 575-595 (1999).