#
The Algebraic Eigenvalue Problem

by numerical-methods.com

The algebraic eigenvalue problem refers to finding a set of
characteristic values associated with a matrix or matrices.
Eigenvalues and eigenvectors are important in that when the
corresponding equations model a physical situation they tell
us useful information about it.
However the eigenvalue problem can take several different forms.
**Traditional Eigenvalue Problem**

The traditional eigenvalue problem involves finding the (eigen)values
l
and the (eigen)vectors __x__ that are non-trivial solutions of

where A is an NxN matrix and __x__ is an N-vector.
Tutorials on Matrix Eigenvalues and Eigenvectors

**Generalised Eigenvalue Problem**

The generalised eigenvalue problem involves finding the (eigen)values
l
and the (eigen)vectors __x__ that are non-trivial solutions of

where A and B are an
N ×N matrix and __x__ is an N-vector. Note that if
B = I, the identity matrix, then this is the traditional
eigenvalue problem.

Tutorials on the Generalised Eigenvalue Problem
**Non-Linear Eigenvalue Problem**

The non-linear eigenvalue problem involves finding the (eigen)values
l
and the (eigen)vectors __x__
that are non-trivial solutions of

where A is an N ×N matrix with each component of A being a
function of the parameter l.
Note that A(l) = C-lI then we have the
traditional eigenvalue problem and if A(l) = C- lD
then we have the generalised eigenvalue problem.

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