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Eigenvalues And Eigenvectors Of A Symmetric Matrix
Syntax eigsym(x)
eigsym(x,e)
See Also symeig , eigen , svd

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 ]
     }