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From: jwunsch@phoenix.Princeton.EDU (Jared Wunsch)
Newsgroups: comp.theory.dynamic-sys,sci.math
Subject: Re: HELP: diagonalizability of nonhermitian matrices theorem
Message-ID: <9575@idunno.Princeton.EDU>
Date: 15 May 91 16:18:42 GMT
References: <1991May13.135859.2737@lgc.com> <91133.162124BJK111@psuvm.psu.edu> <1991May14.102145.5659@bnlux1.bnl.gov>
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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). 

Can this be true?  It seems to me that given an arbitrary matrix A, if we let
A* be its Hermitian transpose, then we have

A = (A+A*)/2 + (A-A*)/2

ie. A is a linear combination of a Hermitian and an anti-Hermitian matrix.  So
if the conjecture were true, we'd have every matrix diagonalizable.