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.