# I Eigenvalue as a generalization of frequency

1. Aug 1, 2016

### npit

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.

2. Aug 1, 2016

### FactChecker

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. Aug 1, 2016

### npit

Thanks.
Did so, I dont understand how it's related to frequency though.
Rotations per time would.

4. Aug 1, 2016

### FactChecker

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. Aug 9, 2016

### chiro

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 occuring 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.