Proving Eigenvector and Eigenvalue Relationship for A and A-cI

In summary, the first step of the proof is to show that Ax=\lambdax, which is what the student tried to do. However, they got stuck on the first line of the proof and needed help from the teacher.
  • #1
maherelharake
261
0

Homework Statement


If v is an eigenvector of A with corresponding eigenvalue [tex]\lambda[/tex] and c is a scalar, show that v is an eigenvector of A-cI with corresponding eigenvalue [tex]\lambda[/tex]-c.


Homework Equations





The Attempt at a Solution


I started out thinking that I have to figure out how to go from:
Ax=[tex]\lambda[/tex]x
to
(A-cI)x=([tex]\lambda[/tex]-c)x

Is this the right start?
 
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  • #2
Looks like a good start to me. Keep going...
 
  • #3
I don't see how I'm supposed to continue. It just seems obvious to me and I can't think of any intermediate steps.
 
  • #4
Try expanding (A-cI) x.
 
  • #5
Ok I expanded the first step to...
Ax-(CI)x=[tex]\lambda[/tex]x-cx
 
  • #6
I also know that CI is similar to cx, except CI is a matrix and cx is not. But I'm stuck...
 
  • #7
I don't see your problem. You showed that [tex] (A-cI)x = (\lambda-c)x[/tex], which is what you were supposed to show.
 
  • #8
No, he didn't show that, he simply wrote it and asked how to go from [itex]Ax= \lamda x[/itex] to that.

maherelharake, From (A- cI)x you get Ax- (cI)x= Ax- c(Ix). Since x is an eigenvector of A corresponding to eigenvalue [itex]\lambda[/itex], [itex]Ax= \lambda x[/itex]. I is the identity operator so Ix= x. Put those together.
 
  • #9
This is what I did step by step. How does it look?
(A-cI)x=([tex]\lambda[/tex]-c)x
Ax-(cI)x=[tex]\lambda[/tex]x-cx
Ax-c(Ix)=[tex]\lambda[/tex]x-cx
Ax-cx=[tex]\lambda[/tex]x-cx
Ax=[tex]\lambda[/tex]x
 
  • #10
Is that sufficient?
 
  • #11
It seems like you are confused with how to go about demonstrating or proving something. You need to start with something you know, then apply known operations until you arrive at what you are trying to prove. Here, the first line of your proof is what you are trying to prove, and you don't yet know it to be true, so if I were grading this proof I would object to the very first line. Actually if you just reverse the order of steps, starting from the bottom and working up, you have pretty well shown what you are trying to prove
 
  • #12
I see what you are saying. So just reverse it?
 
  • #13
Well I guess I will just go with the reverse of what I had...
 

Related to Proving Eigenvector and Eigenvalue Relationship for A and A-cI

1. What is an eigenvalue?

An eigenvalue is a scalar value that represents how a linear transformation changes the direction of a vector. It is a key concept in linear algebra and is often used to analyze and understand the behavior of systems in mathematics, physics, and engineering.

2. How is an eigenvalue calculated?

Eigenvalues are calculated by solving the characteristic equation of a matrix, which is defined as det(A-λI) = 0, where A is the matrix, λ is the eigenvalue, and I is the identity matrix. The solutions to this equation are the eigenvalues of the matrix. The eigenvectors of the matrix can then be found by substituting the eigenvalues back into the equation (A-λI)x = 0 and solving for x.

3. What is the significance of eigenvalues?

Eigenvalues are important because they provide information about the behavior of a linear transformation or matrix. They can indicate whether a system is stable or unstable, and they can be used to find the dominant behavior of a system. Eigenvalues are also used in many applications such as image and signal processing, quantum mechanics, and data analysis.

4. Can the eigenvalues of a matrix be negative?

Yes, the eigenvalues of a matrix can be negative. In fact, a matrix can have both positive and negative eigenvalues, or even imaginary eigenvalues. The sign of the eigenvalues depends on the properties of the matrix, such as its size, symmetry, and determinant.

5. How are eigenvalues related to eigenvectors?

Eigenvalues and eigenvectors are closely related as they are found together when solving the characteristic equation of a matrix. Eigenvectors are the corresponding vectors to the eigenvalues, and they represent the direction in which the linear transformation is applied. The magnitude of the eigenvalue determines the amount of scaling or compression of the eigenvector.

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