At 14:40 04/11/2003, [EMAIL PROTECTED] wrote: ><If I consider an indefinte real matrix ...> >I'm not familiar with the definition of an indefinite matrix.
I see it in Harville, D.A., Matrix algebra from a statistician's perspective. 1997, New-York: Springer-Verlag. 630. it is a matrix A that is neither positive definite nor positive definite. ><I would like to know if the symmetry of the matrix is sufficient to say >that their eigenvectors are real ?> >(I assume you mean the eigenvalues are real...) >If A is a symmetric real valued matrix then the eignevalues of A are >real. There is a short proof of this due to Styan that can be found, >among other places, in Searle's 'Matrix Algebra Useful for Statistics' and >alluded to in Anderson's Multivariate Analysis book in Chapter 11. > >If you truly mean 'do the _eigenvectors_ have to be real?', then not if >you allow A to operate on complex vectors. If A is a real valued >symmetric matrix then it has a real valued eigenvalue, lambda, and a real >valued eigenvector, u which satisfy A * u = lambda * u. If you allow A to >operate on complex vectors then i*u is a complex vector that satisfies >A*(i*u) = lambda*(i*u). Thank you for the answer... I mean eigenvectors are real.... but I agree that it was a real stupid question... As I usually work on p.s.d matrix, I am really suspicious with negative eigenvalues.. >< And what is the conditions to ensure that eigenvectors are real in the >case of an asymmetric matrix (if some conditions exist)?> > >Again, I assume you mean eigenvalues are real, not eigenvectors. I don't >think there is any (useful) condition in general. The eigenvalues are >real if det(A-lambda*I) has real roots. Think of the case of a 2x2 matrix >where you can solve this by the quadratic equation and consider some examples. > Ok, I agree.. so it seems that no useful conditions exist on asymmetric matrices to be sure to obtain real eigenvalues.. St�phane DRAY -------------------------------------------------------------------------------------------------- D�partement des Sciences Biologiques Universit� de Montr�al, C.P. 6128, succursale centre-ville Montr�al, Qu�bec H3C 3J7, Canada Tel : 514 343 6111 poste 1233 E-mail : [EMAIL PROTECTED] -------------------------------------------------------------------------------------------------- Web http://www.steph280.freesurf.fr/ -------------------------------------------------------------------------------------------------- [[alternative HTML version deleted]] ______________________________________________ [EMAIL PROTECTED] mailing list https://www.stat.math.ethz.ch/mailman/listinfo/r-help
