Eigenvalue as a generalization of frequency

In summary, the conversation discusses the concept of eigenvalues and eigenvectors, which can be understood through a geometric intuition where the eigenvectors of a matrix are stretched by the corresponding eigenvalue when transformed through the matrix. The professor mentioned that eigenvalues are a generalization of the concept of frequency, which can be further explained by considering complex eigenvalues and eigenvectors, which represent rotations and expansions in a plane. The conversation suggests checking out spectral decomposition and considering orthogonal matrices to better understand this concept.
  • #1
npit
4
0
Hello everyone.
I understand the concept of eigenvalues and eigenvectors, using usually a geometric intuition, that a eigenvectors of a matrix M are stretched by the corresponding eigenvalue, when transformed through M.

My professor said that eigenvalues represent a generalization of the concept of frequency.
I can not recall the context though.
Can someone provide an short explanation and/or some (not too technical) reading material?

Thanks.
 
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  • #2
Are you allowing complex eigenvalues and eigenvectors? They are rotations & expansions in a plane, which is related to frequency responses. If so, check out .
 
  • #3
Thanks.
Did so, I don't understand how it's related to frequency though.
Rotations per time would.
 
  • #4
Rotations per unit time is the right idea. There are so many different contexts that matrix eigenstructures can be used in that it is hard to do more than give a general intuition. If the (complex) eigenvalue multiplication represents rotation in a unit time, then the amount of rotation in that time (the argument of the eigenvalue) does correspond to a frequency. And the magnitude of the eigenvalue corresponds to a gain (per unit time) at that frequency.

PS. I hate to put words in your professor's mouth. You should probably ask him a follow-up question about what he meant.
 
  • #5
Hey npit,

You might want to consider the spectral decompositon of PDP_inverse in terms of rotations and scalings.

If you have an orthogonal matrix with R*R^t = I [meaning R^t = R_inverse] then you can make sense of a rotation occurring along with a scaling of each axes and then rotating back again.

Co-ordinate system transformations have the same property (like in physical visualization and simulations) and it can help when the P matrices have the PP^t = I property.
 

Related to Eigenvalue as a generalization of frequency

1. What is an eigenvalue?

An eigenvalue is a number associated with a specific matrix that represents the scaling factor of the corresponding eigenvector when multiplied by the matrix. It is a generalization of frequency in the sense that it determines the rate of change of the eigenvector when transformed by the matrix.

2. How is an eigenvalue related to frequency?

Eigenvalues can be thought of as the frequencies of a system, as they represent the rate at which an eigenvector oscillates when transformed by a matrix. This is particularly useful in analyzing dynamical systems and understanding their behavior over time.

3. What is the significance of eigenvalues in physics?

Eigenvalues play a crucial role in many areas of physics, including quantum mechanics, statistical mechanics, and fluid dynamics. In quantum mechanics, they represent the possible energy levels of a system, while in fluid dynamics, they determine the stability of a fluid flow.

4. How do eigenvalues and eigenvectors relate to each other?

Eigenvectors are vectors that remain in the same direction when multiplied by a matrix, and their corresponding eigenvalues determine the magnitude of this transformation. In other words, eigenvectors and eigenvalues are intimately connected, and understanding one helps to understand the other.

5. Can eigenvalues be complex numbers?

Yes, eigenvalues can be complex numbers, and this is particularly relevant in quantum mechanics, where the wave functions of particles are often represented by complex-valued eigenvectors. In such cases, the real part of the eigenvalue represents the frequency, while the imaginary part represents the decay rate of the eigenvector.

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