Invariant Subspace Problem

2011-02-09 9:52 am

a) Let T be an invertible operator with eigenvalue λ. Prove that λ^(-1)
is an eigenvalue of the inverse T^(-1).
b) Let S and T be two commuting linear operators on V (commuting
means ST = TS). Prove that the image space V(S) of S is T-invariant.

回答 (1)

myisland8132

2011-02-09 10:44 pm

✔ 最佳答案

(a) By the information given, Tx = λx. So T-1(Tx) = T-1(λx)

T-1(λx) = x. Sub. x = y/λ

T-1(y) = y/λ

This prove that the eigenvalue of the inverse T-1 is 1/λ

(b) Since ST=TS. If the image space S(V) of S is not T-invariant, then there will have one element x such that it is in S(V) but the image TS(x) is not in S(V).

If this is the case, then TS(x) ∉ S(V).

But TS(x) = ST(x). by the expression of left hand side, we see that the image of TS(x) should be in S(V) (because T(x) ∈ V), which contradicts our previous result. So, S(V) should be T-invariant.

👍 0 👎 0