> Perhaps a more clear explanation of the Schur form is this: if you take a > given 2x2 block on the diagonal of the matrix, call it > > [ T11 T12 ] > [ T21 T22 ] > > then either T21 = 0, or T21 is nonzero. > > If T21 is zero, then T11 and T22 are both real eigenvalues of the original > matrix. > > If T21 is nonzero, then T11 = T22, and T11 +/- sqrt(|T21*T12|) are complex > conjugate eigenvalues. > > So, the subdiagonal element (T21) is guaranteed to be 0 in the case of real > eigenvalues, and nonzero in the case of complex eigenvalues. I will try to > update the documentation to make this a little more clear. > > The rest of A (below the subdiagonal) is not guaranteed to be 0, as far as I > remember.
Thank you very much Patrick, now it is completely clear. Well, actually this is more or less what I have guessed but I was not sure especially about the subdiagonal and the lower part of the matrix. Actually I was thinking that for complex eigenvalues the submatrix was something like: [ u v] [ -v u] à la Cauchy-Riemann but it was a wrong guess... :-) Otherwise only the problem of the orthogonality of Z remains open... maybe it is due to the balancing of the matrix made (maybe) internally by the procedure... In any case, thank you very much for your help! Francesco _______________________________________________ Bug-gsl mailing list [email protected] http://lists.gnu.org/mailman/listinfo/bug-gsl
