Symmetric Eigenproblems

   

The symmetric eigenvalue problem (SEP) is to find the eigenvalues  , , and corresponding eigenvectors , , such that

For the Hermitian eigenvalue problem  we have

For both problems the eigenvalues are real.

When all eigenvalues and eigenvectors have been computed, we write

where is a diagonal matrix whose diagonal elements are the eigenvalues , and Z is an orthogonal (or unitary) matrix whose columns are the eigenvectors. This is the classical spectral factorization   of A.

Two types of driver routines  are provided for symmetric or Hermitian eigenproblems:

• a simple driver (name ending -EV)  , which computes all the eigenvalues and (optionally) the eigenvectors of a symmetric or Hermitian matrix A;
• an expert driver (name ending -EVX)  , which can compute either all or a selected subset of the eigenvalues, and (optionally) the corresponding eigenvectors.

The driver routines are shown in table 3.4. Currently the only simple drivers provided are PSSYEV and PDSYEV.