error in Schur decomposition of a non-symmetric matrix ?

error in Schur decomposition of a non-symmetric matrix ?

[Francesco Abbate]

Hi all,

it seems that I've found an error in GSL manual in the section about
eigenvalues/vectors determination of non-symmetric real matrices. The
manual says:

14.3 Real Nonsymmetric Matrices
===============================

The solution of the real nonsymmetric eigensystem problem for a matrix
A involves computing the Schur decomposition

     A = Z T Z^T

   where Z is an orthogonal matrix of Schur vectors and T, the Schur
form, is quasi upper triangular with diagonal 1-by-1 blocks which are
real eigenvalues of A, and diagonal 2-by-2 blocks whose eigenvalues are
complex conjugate eigenvalues of A. The algorithm used is the
double-shift Francis method.
-----------------------

and after about the function gsl_eigen_nonsymmv and gsl_eigen_nonsymmv_Z :

-- Function: int gsl_eigen_nonsymmv (gsl_matrix * A,
          gsl_vector_complex * EVAL, gsl_matrix_complex * EVEC,
          gsl_eigen_nonsymmv_workspace * W)
     This function computes eigenvalues and right eigenvectors of the
     N-by-N real nonsymmetric matrix A. It first calls
     `gsl_eigen_nonsymm' to compute the eigenvalues, Schur form T, and
     Schur vectors. Then it finds eigenvectors of T and backtransforms
     them using the Schur vectors. The Schur vectors are destroyed in
     the process, but can be saved by using `gsl_eigen_nonsymmv_Z'. The
     computed eigenvectors are normalized to have unit magnitude. On
     output, the upper portion of A contains the Schur form T. If
     `gsl_eigen_nonsymm' fails, no eigenvectors are computed, and an
     error code is returned.

 -- Function: int gsl_eigen_nonsymmv_Z (gsl_matrix * A,
          gsl_vector_complex * EVAL, gsl_matrix_complex * EVEC,
          gsl_matrix * Z, gsl_eigen_nonsymmv_workspace * W)
     This function is identical to `gsl_eigen_nonsymmv' except that it
     also saves the Schur vectors into Z.
------------------------------

The problems seems to be about the Schur decomposition formula. What
it is true is that:

A = Z T Z^(-1)

and not that: A = Z T Z^T as asserted in the documentation. I've found
that empirically.

Otherwise I'm perplexed about the statement in the gsl_eigen_nonsymmv:
"On output, the upper portion of A contains the Schur form T". I don't
understand this statement because before it was stated that: "T, the
Schur form, is quasi upper triangular with diagonal 1-by-1 blocks
which are real eigenvalues of A, and diagonal 2-by-2 blocks whose
eigenvalues are complex conjugate eigenvalues of A". So the T matrix
is not really upper triangular.

I've made some tests and it seems that the function gsl_eigen_nonsymmv
set the matrix A to the Schur form T and not only the upper part.

It would be nice if someone could check because what I've seen is
based on empirical tests using the functions.

Best regards,
Francesco