Xref: utzoo comp.theory.dynamic-sys:254 sci.math:17409 Path: utzoo!utgpu!news-server.csri.toronto.edu!rpi!zaphod.mps.ohio-state.edu!swrinde!cs.utexas.edu!sun-barr!ccut!wnoc-tyo-news!astemgw!kuis!kurims!mcanally From: mcanally@kurims.kyoto-u.ac.jp (David Scott McAnally) Newsgroups: comp.theory.dynamic-sys,sci.math Subject: Re: HELP: diagonalizability of nonhermitian matrices theorem Message-ID: Date: 15 May 91 06:57:49 GMT References: <1991May13.135859.2737@lgc.com> <91133.162124BJK111@psuvm.psu.edu> <1991May14.102145.5659@bnlux1.bnl.gov> Sender: bnews@kurims.kyoto-u.JUNET Organization: Research Institute for Mathematical Sciences, Kyoto, Japan. Lines: 28 In-reply-to: kyee@bnlux1.bnl.gov's message of 14 May 91 10:21:45 GMT In article <1991May14.102145.5659@bnlux1.bnl.gov> kyee@bnlux1.bnl.gov (kenton yee) writes: >hi, >we know any hermitian matrix H and any antihermitian matrix >A is diagonalizable. i want to prove that any linear combination >M = aH + bA are diagonalizable (a,b real). one way is to >note that M' = aH + ibA is hermitian and diagonalizable, >so one can continue M' to M by taking b->-ib. this should >be ok unless one of the eigenvectors has a singularity blocking >the analytic continuation. does anyone have a better proof? >also, is there a relation between the eigenvectors of H, A >and M? > >thanks, --ken Surely, this requires specific knowledge of the matrices involved, eg, the (hermitian) Pauli matrices sigma , sigma and sigma are 1 2 3 diagonalizable, but sigma +/- i sigma are not. 1 2 If H and A commute, then there are no problems with diagonalizability. David McAnally kurims.kurims.kyoto-u.ac.jp ``Other kings said I was daft to build a castle on a swamp." King of Swamp Castle: Monty Python and the Holy Grail