Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Distribution of 2 matrices with the same eigenvalues

  1. Nov 24, 2015 #1

    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


    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    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

    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


    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    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.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook