Motivations for eigenvalues/vectors

  • Thread starter eckiller
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In summary, the motivation for eigenvalues/vectors is to compute A^k * x. This can be done by writing A as C*D*C^-1, where A^k = C * D^k * C^-1 and D^k is easy to compute. C^-1 and C can be thought of as change of coordinate matrices, where C^-1 takes vectors from a given basis to the basis of eigenvectors and C takes them back. The given basis is the basis in which A is written, and the eigenvector basis is the one consisting of eigenvectors for A. A matrix can only be diagonalized if there exists a basis consisting entirely of its eigenvectors. This basis is not explicitly stated
  • #1
eckiller
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Hi,

I understand one of the motivations for eigenvalues/vectors is when you need
to compute A^k * x. So we like to write,

A = C*D*C^-1 and then A^k = C * D^k * C^-1, and D^k is trivial to compute.

My professor said C^-1 and C can be though of as change of coordinate
matrices. But from which basis? For example, C^-1 would take me from
*some* basis to the basis of eigenvectors. But what is this *some* basis?

Is it assumed that everything is coordinitized relative to some basis B in
R^n. And then I want to change to the basis of eigenvectors B'?
 
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  • #2
Short answer: Yes.

A is the matrix of a transformation wrt some basis B. D is the matrix of the same transformation wrt the eigenbasis B'. [itex]C^{-1}[/itex] takes vectors from B to B', and C takes vectors back from B' to B.
 
  • #3
A general linear transformation can be written as a matrix in a given basis. If all you are given is the matrix, then the corresponding "given basis" is <1, 0,...>, <0,1,...> , etc. The basis you are changing to is the basis consisting of the eigenvectors for the matrix A.

(A matrix can be diagonalized if and only if there exist a basis consisting entirely of eigenvectors of the matrix.)
 
  • #4
Thanks for clearing that up. I wish both of my linear algebra textbooks made it clear which basis we were in.
 
  • #5
But it obviously goes from whatever basis A is written with respect to, and it doesn't matter what that basis is, which is why the book didn't state it.
 

1. What are eigenvalues and eigenvectors?

Eigenvalues and eigenvectors are concepts in linear algebra that are used to describe the behavior of a linear transformation on a vector space. Eigenvalues represent the scaling factor of the eigenvector when it is transformed by the linear transformation.

2. Why are eigenvalues and eigenvectors important?

Eigenvalues and eigenvectors are important because they provide valuable information about the properties of a linear transformation. They can be used to analyze the behavior and stability of a system, and are also essential in many applications in physics and engineering.

3. What are some motivations for studying eigenvalues and eigenvectors?

One motivation for studying eigenvalues and eigenvectors is their usefulness in solving systems of linear equations. They can also be used to describe the behavior of dynamical systems, such as in population dynamics or chemical reactions. Additionally, eigenvalues and eigenvectors are important in fields like quantum mechanics, computer graphics, and signal processing.

4. How do you calculate eigenvalues and eigenvectors?

The process of finding eigenvalues and eigenvectors involves solving a system of linear equations. This can be done by finding the characteristic polynomial of the matrix, setting it equal to zero, and solving for the roots. The eigenvectors can then be found by plugging in the eigenvalues into the system of equations.

5. Can eigenvalues and eigenvectors have complex values?

Yes, eigenvalues and eigenvectors can have complex values. In fact, in some cases, complex eigenvalues can provide more information about the system than real eigenvalues. Complex eigenvectors are also commonly used in applications such as quantum mechanics.

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