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Eigenvalues And Eigenvectors Of A Symmetric Matrix

Description
Computes the eigenvalues of the matrix x, where x is a real, double-precision or complex conjugate symmetric matrix .

If the argument e is present the eigenvectors of x are also computed. The input value of e does not matter and its output value is a unitary matrix , with the same type and dimension as x, containing the eigenvectors.

The return value is a column vector with the same type and row dimension as x containing the eigenvalues. If r is the return value,
e * diag(r) = x * e
where diag(r) is the diagonal matrix with r along its diagonal.

This routine computes the eigenvalues and eigenvectors directly. The routine symeig uses an SVD factorization to compute the eigenvectors.

Example
If you enter
x  = {[3., 1.], [1., 3.]}
e  = novalue
eigsym(x, e)
O-Matrix will respond
{
2
4
}
If you continue by entering
e * e'
O-Matrix will respond
{
[ 1 , 0 ]
[ 0 , 1 ]
}
(Note that the complex conjugate is not necessary because e has the same type as x and hence is a real matrix.) If you continue by entering
e * diag({2, 4}) - x * e
O-Matrix will respond
{
[ 0 , 0 ]
[ 0 , 0 ]
}