The symmetric eigenvalue problem 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.
Three types of driver routines are provided for symmetric or Hermitian eigenproblems:
Different driver routines are provided to take advantage of special structure or storage of the matrix A, as shown in Table 2.5.
In the future LAPACK will include routines based on the Jacobi algorithm , which are slower than the above routines but can be significantly more accurate.