# Eigenvalues/vectors question

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#1
OK, I've got a 2-parter today...

Let T be a linear transformation T : V -> V , where V is a finite-dimensional
vector space over R.

i. Show that if T is invertible, then the eigenvalues of T^(-1) are the reciprocals of the eigenvalues of T? What can be said about the corresponding eigenvectors?

ii. Show that if v is an eigenvector of T associated with the eigenvalue a and
k is any natural number, then v is an eigenvector of T k associated with the
eigenvalue a?

Any ideas for either part?
0
15 years ago
#2
(Original post by flyinghorse)
OK, I've got a 2-parter today...

Let T be a linear transformation T : V -> V , where V is a finite-dimensional
vector space over R.

i. Show that if T is invertible, then the eigenvalues of T^(-1) are the reciprocals of the eigenvalues of T? What can be said about the corresponding eigenvectors?

ii. Show that if v is an eigenvector of T associated with the eigenvalue a and
k is any natural number, then v is an eigenvector of T k associated with the
eigenvalue a?

Any ideas for either part?
i. Suppose that Tv = kv. Then

v = T^{-1}(kv) = k T^{-1}v

and so

T^{-1}v = (1/k)v.

So the eigenvalues of T^{-1} are the reciprocals of the eigenvalues of T and the eigenvectors are the same.

ii. Don't understand your question here - what does "T k" mean, maybe "T^k"? In which case v is an eigenvector of T^k but has eigenvalue a^k.
0
#3
(Original post by RichE)
i. Suppose that Tv = kv. Then

v = T^{-1}(kv) = k T^{-1}v

and so

T^{-1}v = (1/k)v.

So the eigenvalues of T^{-1} are the reciprocals of the eigenvalues of T and the eigenvectors are the same.

ii. Don't understand your question here - what does "T k" mean, maybe "T^k"? In which case v is an eigenvector of T^k but has eigenvalue a^k.
oops, sorry - you're right, it's supposed to say T^k.

cheers
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