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Distribution of 2 matrices with the same eigenvalues

  1. Nov 24, 2015 #1
    Hi,

    I was wondering if two matrices with the same eigenvalues share the same PDF.

    Any ideas and/or references would be helpful.
    Thanks in advance
     
  2. jcsd
  3. Nov 24, 2015 #2

    andrewkirk

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    You'll have to provide some more information about what you're seeking.
    Matrices in general don't have anything that is widely referred to as a 'PDF'.
    The only PDF I know is 'probability density function' in probability theory. Is that what you mean? If so, how do you want to relate it to a matrix? Matrices are used in probability theory and statistics in numerous different ways.
     
  4. Nov 25, 2015 #3
    Hello,

    Assume that H is a n \times m matrix with i.i.d. complex Gaussian entries each with zero mean and variance \sigma. Also, let n>=m. I ' m interested in finding the relation between the distribution of HHH and HHH, where H stands for the Hermittian transposition. I anticipate that both follow the complex wishart distribution with the same parameters (since they share the same nonzero eigenvalues), but I m not sure about this.

    Any ideas ? Thanks in advance..
     
  5. Nov 25, 2015 #4

    andrewkirk

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    The matrix ##H^HH## will be ##m\times m## while ##HH^H## will be ##n\times n##. They will have different numbers of eigenvalues. Why do you think the nonzero ones will be the same?
     
  6. Nov 25, 2015 #5
    Indeed, they have different dimensions. However their non-zero eigenvalues are the same. This is a fact. If you hold reservations about the latter just implement it in Matlab and see with the command eig their corresponding eigenvalues.

    My question is: if they also have the same probability density function.
     
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