Eigenvalue as a generalization of frequency

[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.

 

[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 .

 

[npit]

Thanks.
Did so, I don't understand how it's related to frequency though.
Rotations per time would.

 

[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.

 

[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 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.