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  <jean-claude.serlet@insee.fr> 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