SOLUTION: Suppose V is an n dimensional vector space and that T is an operator on V. T has n distinct eigenvalues. Also suppose that S is also an operator on V which has the same eigenvector

Question 15391: Suppose V is an n dimensional vector space and that T is an operator on V. T has n distinct eigenvalues. Also suppose that S is also an operator on V which has the same eigenvectors as T ( not necessarily with the same eigenvalues).
Prove that ST = TS
Answer by khwang(438) (Show Source): You can put this solution on YOUR website!
Suppose V is an n dimensional vector space and that T is an operator on V. T has n distinct eigenvalues. Also suppose that S is also an operator on V which has the same eigenvectors as T ( not necessarily with the same eigenvalues).
Prove that ST = TS
Proof: If T has n distinct eigenvalues {λi| i=1,2,..,n}(i.e. ) with corresponding eigenvectors {vi|i=1,2,..n},
then {vi|i=1,2,..n} is linear independent and so forms a basis of V.
For each i, vi is an eigenvector of S means there exists scalar βi
such that Svi = βi vi.
Consider ST (vi) = S(λi vi ) = λi S(vi) = λiβi vi and
TS (vi ) = TS (βi vi ) = βi T(vi) =λiβi vi for every i=1,2,..n.
This shows ST = TS (since {vi|i=1,2,..n} is a basis of V).
Fact. The eigenvectors {vi|i=1,2,..,n} of T is linearly independent
Proof: If for some scalars ci,
set L to be the product operator
II(T �λj I) (j=1,2,..,n-1 ). Then, we have L(Σci vi) =Σci L(vi)
= Σci (λi vi -λi vi) (i=1,2,..,n-1) + cn II(λn -λi )vn
= cn II(λn -λi )vn = 0 .
Since all {λi} are distinct, II(λn -λi ) <> 0. (i=1,2,..,n-1)
We obtain cn = 0. Similarly, we can prove that all other scalars ci are zeros. This shows that {vi} are independent.
Kenny
PS. I don't quite believe that some questions like this would be posted
here.
RELATED QUESTIONS
Prove that a linear operator T on a finite-dimensional vector space is invertible if and... (answered by khwang)

Let T be a linear operator on a finite-dimensional vector space V, and let b (beta) be... (answered by khwang)

suppose that V is finite-dimensional space and T:V->V is a linear operator. 1. prove... (answered by ikleyn)

For the following linear operator T on the vector space V, test T for diagonalizability,... (answered by khwang)

Let u, v, and w be distinct vectors of a vector space V. Show that if {u, v, w} is a... (answered by khwang)

Let "T" be a linear Operator on a finite dimensional space "V" and let "c" be a Scalar. (answered by khwang)

Suppose S,T are subspaces of V and s intersect T=0. Show that every vector in S + T can... (answered by venugopalramana)

Its several months before Christmas - New Year’s holiday season. Wholesalers for... (answered by ikleyn)

This is my second and last question. This is not in a textbook. a.)Let T be a linear... (answered by venugopalramana)