On Wed, 05 Dec 2012 16:11:21 +0000, José Carlos Santos <firstname.lastname@example.org> wrote:
>Hi all, > >A classical Linear Algebra exercise says: prove that _n_ eigenvectors >of an endomorphism of some linear space with _n_ distinct eigenvalues >are linear independent. Does this hold for modules over division rings?
I believe so - I don't see where the standard proof uses commutattivity, or however one spells it.
Say x_1, ... x_n are eigenvectors wrt disctinct eigenvalues l_j. Say
sum c_j x_j = 0.
Applying that endomorphism k times shows that
sum c_j l_j^k x_j = 0
for k = 0, 1, ... . Hence
sum c_j P(l_j) x_j = 0
for any polynomial P.
Now let P be the obvious product, so that P(l_1) <> 0 while P(l_j) = 0 for j >= 2. Then
c_1 P(l_1) x_1 = 0;
since it's a division ring and x_1 <> 0 this shows that c_1 = 0.