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I Eigenvalue as a generalization of frequency

  1. Aug 1, 2016 #1
    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?

  2. jcsd
  3. Aug 1, 2016 #2


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    Are you allowing complex eigenvalues and eigenvectors? They are rotations & expansions in a plane, which is related to frequency responses. If so, check out .
  4. Aug 1, 2016 #3
    Did so, I dont understand how it's related to frequency though.
    Rotations per time would.
  5. Aug 1, 2016 #4


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    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.
  6. Aug 9, 2016 #5


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    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.
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