There are two covariance matrices $U$ and $V$ such that $U= X^\top X$ and $V = XX^\top$ where $X$ is a $d\times N$ matrix. How can I prove that $U$ and $V$ have the same non-zero eigenvalues?

  • $\begingroup$ You mean, nonzero eigenvalues. $\endgroup$
    – Igor Rivin
    Jan 9, 2014 at 0:44
  • $\begingroup$ Duplicate of this question: math.stackexchange.com/questions/30072/… $\endgroup$
    – Igor Rivin
    Jan 9, 2014 at 0:50
  • $\begingroup$ thanks for the reply. I went through the other question but found it very difficult for me. I am a beginner. Can u please explain in simple terms? $\endgroup$ Jan 9, 2014 at 1:39

1 Answer 1


This should really be a comment- but I have too low reputation to make a comment. Hence I am writing it as an answer.

It's a very standard theorem in liner algebra that for two matrices A and B, AB and BA have the same non zero eigenvalues. Take A=$X^T$, and B= X to get your result.


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