From: rusin@vesuvius.math.niu.edu (Dave Rusin) Subject: Re: endomorphism Date: 6 Jan 2000 18:26:38 GMT Newsgroups: sci.math Summary: trace of endomorphism disallows it from being commutator In article <02a4ec93.8f6f78b6@usw-ex0110-075.remarq.com>, ebru2w wrote: > U and V 2 endomorphism ( V,U:E--->E,E is one EV with >dimE=2) ,U different from 0,UoV-VoU=U. > You have to show that U is not one bijection et give >dimKerU Let's see, U and V are 2 endomorphisms (that is, V, U : E --> E where E is a vector space with dim(E)=2 ), U is different from 0, but the difference of the compositions U o V - V o U is equal to U. I have to (a) show U is not a bijection (b) compute dim(ker(U)) Of course (b) is easy given (a), since ker(U) is a subspace of E, which is 2-dimensional, and we have assumed ker(U) is a _proper_ subspace of E while (a) shows it's not the _trivial_ subspace. So dim(ker(U)) will have to be 1. Thus we need only complete (a). The problem here is that what is to be proved isn't really true! Over a field of characteristic 2, take U = [[1,1],[0,1]] and V = [[1,0],[1,0]]. Then UV-VU = U ! So we will have to abandon purely geometric or axiomatic approaches and use something which uses algebra. Note that trace(UV)=trace(VU) so trace(U) will have to be zero, and thus its characteristic roots are negatives of each other AND THUS distinct. But then (passing to vector spaces over an extension fied as necessary) a basis for E may be chosen so that U is represented by a diagonal matrix. A simple computation then shows UV-VU will have both diagonal entries be zero, so that UV-VU = U forces U=0, a contradiction. Of course you could also simply chug out the equation UV-VU=U as a set of 4 equations in 8 unknowns and then find a linear combination of them which forces the first four to vanish. This looks a little magical rather than informative, but indeed we find trace(adj(U)*(UV-VU-U)) = (-2)*det(U), so the premises of the problem show that det(U)=0 when 2 does not equal zero. dave