# On computing the eigenvectors of a class of structured matrices

## Nicola Mastronardi

A real symmetric matrix of order n has a full set of orthogonal eigenvectors.
The approach often used to solve the eigenproblem reduces the dense symmetric
matrix first into a symmetric structured one, i.e., a tridiagonal matrix or a
semiseparable matrix. This step is accomplished in *O(n ^{3})* operations. Once the
latter symmetric structured matrix is available, the eigenvalues of the latter
matrix are computed in an iterative fashion by means of the QR method in

*O(n*operations.

^{2})*O(n*operations.

^{2})
The problem in this approach is that the computed eigenvectors may not be
numerically orthogonal if clusters are present in the spectrum. To enforce
orthogonality the Gram-Schmidt procedure is used, requiring *O(n ^{3})* operations
in the worst case.

In this talk we consider a new fast and stable method (*O(n ^{2})* operations) to
compute all the eigenvectors of tridiagonal and semiseparable matrices
which does not suffer from the presence of clusters in the spectrum. Also the
problem of computing the eigenvectors of unsymmetric matrices is handled.