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

A x = lx
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

A x = lB x
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

A(l) x = 0
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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