From: kovarik@mcmail.cis.McMaster.CA (Zdislav V. Kovarik) Subject: Re: generalized eigenvalues with ranck deficiency Date: 1 Feb 2000 11:14:49 -0500 Newsgroups: sci.math Summary: [missing] In article <3896F68C.2A8FC9FA@unina.it>, Ciro Caramiello wrote: :Let us suppose that we want to solve this generalized eigenvalues :problem: A*z=lambda*B*z where A and B are two (n*n) complex matrices in You probably meant "two (m*n) complex matrices; the case (n*n) with A non-singular is exactly as hard as the case A=I (just replace B with A^(-1)*B and A with I) :particular A is full-rank while B has one or more null (all zeros) rows. : :I'd like to know more algebraic details on this problems since the :generalized spectrum has n-m eigenvalues and, consequently, n-m :eigenvectors (to prove this apply directly the definition of :characteristic polynomial) where m is the number of zero rows. Moreover :I'd like to get more infos about the QZ algorithm which proved to find :correctly the n-m eigenvalues but, of course, assigns infinity values to : :the m "phantom" eigenvalues. What about the eigenvectors in particular :when A is quasi rank deficient? Please contact me if you think you could : : help me to solve this non-classical problem of linear algebra and :numerical analysis. It is actually classical, provided one considers Kronecker a classic. Look up "matrix pencils" and "Kronecker normal form" in Gantmakher's or Lancaster's treatise, and a numerical discussion was done by John Wilkinson in the early 80's of last century. I am afraid it exists only in a journal article form. (I'd be grateful for a correction if I am wrong here.) Good luck, ZVK(Slavek).