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eig
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1995-02-12
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eig:
Syntax: eig ( A )
eig ( A , B )
eigs ( A )
eigs ( A , B )
eign ( A )
eign ( A , B )
Description:
eig ( A )
Computes the eigenvectors, and values of matrix
A. eig() returns a LIST with elements `val' and `vec'
which are the eigenvalues and eigenvectors. Eig checks
for symmetry in A, and uses the appropriate solver.
eig ( A , B )
Computes the eigenvectors, and values of A, and B.
Where A * x = lambda * B * x. The values and vectors
are returned in a list with element names `val' and
`vec'. Eig checks for symmetry in A and B and uses the
appropriate solver.
eigs ( A )
This function solves the standard eigenvalue problem,
like eig, but the symmetric solver is always
used. Returns a list with elements `val' and `vec'.
eigs ( A , B )
The symmetric solution to the generalized eigenvalue
problem. Returns a list with elements `val' and
`vec'. Always uses the symmetric eigensolver.
eign ( A )
This function solves the standard eigenvalue problem,
like eig, but the non-symmetric solvers are always
used. eign returns a list containing:
lvec: A matrix of the left eigenvectors.
rvec: A matrix of the right eigenvectors.
val: A row vector of the eigenvalues.
eign ( A , B )
The nonsymmetric solution to the generalized
eigenvalue problem. Returns a list containing:
alpha: A row vector, such that val = alpha / beta
beta: A row vector, such that val = alpha / beta
lvec: A matrix of the left eigenvectors.
rvec: A matrix of the right eigenvectors.
Uses the LAPACK subroutines DSYEV/ZHEEV or DGEEV/ZGEEV.
Example:
The generalized eigenvalue problem arises quite regularly in
structures. From the second order differential equations
describing a lumped mass system arise $M$ and $K$, coefficient
matrices representing the mass and stiffness of the various
physical degress of freedom. The equations are formulated as
follows:
dx^2
M --- + K x = F
dt^2
Which leads to the eigenvalue problem:
K v = w^2 M v
For a two degree of freedom system we might have:
> m = eye(2,2)
> k = [5,1;1,5]
> </ val ; vec /> = eig(k, m);
> // Test the solution
> k * vec[;1]
-2.83
2.83
> val[1] * m * vec[;1]
-2.83
2.83
> // Properties of the solution
> vec' * m * vec
1 -4.27e-17
-4.27e-17 1
> vec' * k * vec
4 -1.71e-16
1.23e-16 6
The eigenvalues and vectors are sometimes obtained by
converting the generalized problem into a standard eigenvalue
problem (this is only feasible under certain conditions).
> a = m\k
a =
5 1
1 5
> eig(a).val
val =
4 6
> eig(a).vec
vec =
-0.707 0.707
0.707 0.707
See Also: svd, schur