• Support PF! Buy your school textbooks, materials and every day products Here!

Tough Eigenvalue problem

  • Thread starter temaire
  • Start date
  • #1
279
0

Homework Statement


[PLAIN]http://img28.imageshack.us/img28/5227/79425145.jpg [Broken]


The Attempt at a Solution



I'm not exactly sure how to go about this problem. How do I start?
 
Last edited by a moderator:

Answers and Replies

  • #2
Dick
Science Advisor
Homework Helper
26,258
618


Start with A^2. A^2(x)=A(A(x)). What's that in terms of lambda?
 
  • #3
279
0


Is it lambda^2(x)
 
  • #4
Dick
Science Advisor
Homework Helper
26,258
618


Is it lambda^2(x)
Sure. So that means x is an eigenvector of A^2 with eigenvalue lambda^2, right? The statement for general N>0 follows in the same way.
 
  • #5
279
0


So the solution is simply:
Ax = lambda x
Therefore A^n x = lambda^n x ?

Is there something I'm missing?
 
  • #6
Dick
Science Advisor
Homework Helper
26,258
618


So the solution is simply:
Ax = lambda x
Therefore A^n x = lambda^n x ?

Is there something I'm missing?
That's what the problem is asking you to show, isn't it?
 
  • #7
279
0


That's what the problem is asking you to show, isn't it?
So what you showed me with A^2 is all I need to answer the question?
 
  • #8
Dick
Science Advisor
Homework Helper
26,258
618


Yes it is, but is that all there is to it?
If you understand why it's true, then yes, that's all there is to it. If you want to be formal about proving it you might want to present it as an induction proof.
 
  • #9
279
0


Here is my solution:

A^2(x) = A(A(x))
A^2(x) = lambda^2(x)

Therefore x is an eigenvector of A^2 with eigenvalue lambda^2. The general statement for n>0 follows in the same way.

Is this complete?
 
  • #10
33,154
4,838


You should probably use induction rather than say "The general statement for n>0 follows in the same way." That's very vague.
 
  • #11
279
0


You should probably use induction rather than say "The general statement for n>0 follows in the same way." That's very vague.
What do you mean by induction? I've never learned it.
 
  • #13
Dick
Science Advisor
Homework Helper
26,258
618


What do you mean by induction? I've never learned it.
Induction is the formal way to show it. If you've never heard of it and aren't expected to use it the alternative is 'hand waving'. It's easy enough to prove A^3(x)=lambda^3 x using A^2(x)=lambda^2 x. That makes it easy to show A^4(x)=lambda^4 x, etc, etc. So we see it's true for all n. Induction is just the formal way to state 'etc etc'. It's up to you to decide whether the course requires it.
 

Related Threads for: Tough Eigenvalue problem

Replies
6
Views
839
  • Last Post
Replies
12
Views
2K
  • Last Post
Replies
9
Views
1K
  • Last Post
Replies
13
Views
1K
  • Last Post
Replies
10
Views
2K
  • Last Post
Replies
3
Views
1K
  • Last Post
Replies
0
Views
1K
  • Last Post
Replies
8
Views
1K
Top