Eigenvalues of nonsymmetric matrices play an essential role in physical
problems. These eigenvalues may be used either for their physical meaning (vibration
modes, stability region,...) or for their numerical meaning (convergence of
numerical scheme,...).
However, the eigenproblem to be solved becomes more and more difficult
as the matrix under study gets significantly far from normal (
)
[6].
And highly nonnormal problems are encountered more and more
frequently in physics.
Such nonnormal matrices have unstable spectra so that
the computed eigenvalues can be far away from the exact ones.
It is therefore important to analyse the spectral behaviour under data
perturbations.
Perturbations on the computed eigenvalues are the consequence
of perturbations 
on the matrix A. These perturbations may come from
physics (uncertainty on the data,...) and/or from numerics (numerical
approximations, finite precision arithmetic).
For isolated eigenvalues, the condition number is usually sufficient
to predict the influence of a perturbation on the
matrix. For defective eigenvalues,
no exact formulation for the condition number is available. And moreover, in
presence of nonnormality, separated eigenvalues in exact arithmetic
tend to be coupled by the finite precision computation.
This is why there is a need to develop global tools to investigate
the influence of perturbations on the spectrum
of nonnormal matrices (see [6], [10], [13],
[14], [12]).
One way to carry out such a study is to compute the spectral portrait of
matrix that we describe in the next section.