Math Help - Eigenvectors/values Question

1. Eigenvectors/values Question

(a) [8 marks] Let V be a real vector space and let $T: V\rightarrow V$ be a linear transformation.
Define the terms eigenvalue and eigenvector of T. Show that if $v_1, . . . , v_n$ are eigenvectors
for distinct eigenvalues $\lambda_1,...,\lambda_n$ then the set of $v_1,...,v_n$ is linearly independent.

(b) [12 marks] Now let $v_1,...,v_n$ be a basis of eigenvectors with distinct eigenvalues $\lambda_1,...,\lambda_n$. Prove that for any linear map $S:V\rightarrow V$ with ST = TS, $v_i$ 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?

2. Re: Eigenvectors/values Question

Originally Posted by worc3247
(a) [8 marks] Let V be a real vector space and let $T: V\rightarrow V$ be a linear transformation.
Define the terms eigenvalue and eigenvector of T. Show that if $v_1, . . . , v_n$ are eigenvectors
for distinct eigenvalues $\lambda_1,...,\lambda_n$ then the set of $v_1,...,v_n$ is linearly independent.

(b) [12 marks] Now let $v_1,...,v_n$ be a basis of eigenvectors with distinct eigenvalues $\lambda_1,...,\lambda_n$. Prove that for any linear map $S:V\rightarrow V$ with ST = TS, $v_i$ 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?

Hint for (a): use induction.

For (b): use this this theorem:

Matrix M_n is similar to diagonal Matrix D iff there are exist n linear independent eigenvectors, diagonal elements of D are eigenvalues and D=P^{-1}MP , where P is matrix with eigenvectors on the columns.

3. Re: Eigenvectors/values Question

Originally Posted by worc3247
(a) [8 marks] Let V be a real vector space and let $T: V\rightarrow V$ be a linear transformation.
Define the terms eigenvalue and eigenvector of T. Show that if $v_1, . . . , v_n$ are eigenvectors
for distinct eigenvalues $\lambda_1,...,\lambda_n$ then the set of $v_1,...,v_n$ is linearly independent.

(b) [12 marks] Now let $v_1,...,v_n$ be a basis of eigenvectors with distinct eigenvalues $\lambda_1,...,\lambda_n$. Prove that for any linear map $S:V\rightarrow V$ with ST = TS, $v_i$ 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?
If these questions are for marks, it is against forum policy knowingly to help with such questions.

Thread closed. PM me if you wish to discuss it further.

4. Re: Eigenvectors/values Question

Thread re-opened based on PM from OP'er.