<If I consider an indefinte real matrix ...> I'm not familiar with the definition of an indefinite matrix.
<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). < 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. Bob -----Original Message----- From: Stephane DRAY [mailto:[EMAIL PROTECTED] Sent: Tuesday, November 04, 2003 1:14 PM To: R help list Subject: [R] real eigenvectors Hello list, Sorry, these questions are not directly linked to R. If I consider an indefinte real matrix, I would like to know if the symmetry of the matrix is sufficient to say that their eigenvectors are real ? And what is the conditions to ensure that eigenvectors are real in the case of an asymmetric matrix (if some conditions exist)? Thanks in Advance, 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/ ______________________________________________ [EMAIL PROTECTED] mailing list https://www.stat.math.ethz.ch/mailman/listinfo/r-help [[alternative HTML version deleted]] ______________________________________________ [EMAIL PROTECTED] mailing list https://www.stat.math.ethz.ch/mailman/listinfo/r-help
