Reference no: EM131025572

Assume that ST = TS. Prove that the operators S and T have a common eigenvector.

Let V be a complex (i.e. F = R) finite dimensional vector space. Let S, T be elements of
L(V ) (set of operators on V).

Assume that ST = TS.

Prove that the operators S and T have a common eigenvector.
these are the steps:

a) Explain why T has at least one eigenvalue. We will denote this eigenvalue by λ.

Let W be the set of all eigenvectors of T which have the eigenvalue λ, together with the zero vector, 0 element of V .

b) Prove that W is a subspace of V .

c) Prove that W is invariant under S.

d) Prove that there is an eigenvector for S which belongs to W.

e) Finish the proof of the claimed result.