(a) [8 marks] Let V be a real vector space and let
be a linear transformation.
Define the terms eigenvalue and eigenvector of T. Show that if
are eigenvectors
for distinct eigenvalues
then the set of
is linearly independent.
(b) [12 marks] Now let
be a basis of eigenvectors with distinct eigenvalues
. Prove that for any linear map
with ST = TS,
is also an eigenvector of S for each i = 1, . . . , n.
Deduce that an n×n matrix that commutes with every diagonal n×n matrix must
be diagonal itself.
So far I have managed part a), but am stuck on the first part of b). I have tried taking TS(v) and then using the eigenvalues and the fact that S and T are linear to find something, but I can't get anything of the form S(v)=av (a constant). Could someone start me off?