Tough Eigenvalue problem

In summary, the conversation discusses how to solve a problem involving lambda and A^2. The solution is shown to involve using induction, which is formally proven in the conversation. It is concluded that the solution is complete and the problem is solved.
  • #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:
Physics news on Phys.org
  • #2


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


Is it lambda^2(x)
 
  • #4


temaire said:
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


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

Is there something I'm missing?
 
  • #6


temaire said:
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


Dick said:
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


temaire said:
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


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


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


Mark44 said:
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


temaire said:
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.
 

1. What is a "Tough Eigenvalue problem"?

A Tough Eigenvalue problem is a mathematical problem that involves finding the eigenvalues and eigenvectors of a large and complex matrix. It is considered tough because it requires sophisticated numerical methods and computational resources to solve.

2. What is the importance of solving Tough Eigenvalue problems?

Solving Tough Eigenvalue problems is important in various fields such as physics, engineering, and computer science. It allows researchers and scientists to understand complex systems and make predictions about their behavior. It also has applications in machine learning, image processing, and quantum mechanics.

3. What are some common methods used to solve Tough Eigenvalue problems?

Some common methods used to solve Tough Eigenvalue problems include the Power Method, Inverse Power Method, Jacobi Method, and QR Algorithm. These methods use different approaches and techniques to find the eigenvalues and eigenvectors of a matrix.

4. Are there any challenges or limitations when solving Tough Eigenvalue problems?

Yes, there are several challenges and limitations when solving Tough Eigenvalue problems. These include the size and complexity of the matrix, numerical instability, and the potential for multiple or complex eigenvalues. Additionally, the computational resources and time required to solve these problems can be significant.

5. How can I learn more about solving Tough Eigenvalue problems?

There are various resources available for learning about solving Tough Eigenvalue problems. These include textbooks, online courses, and research papers. It is also helpful to have a strong understanding of linear algebra and numerical methods. Consulting with experts in the field or joining a research group can also provide valuable insights and guidance.

Suggested for: Tough Eigenvalue problem

Back
Top