# Prove two covariance matrices have same eigen values?

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?

• You mean, nonzero eigenvalues. – Igor Rivin Jan 9 '14 at 0:44
• Duplicate of this question: math.stackexchange.com/questions/30072/… – Igor Rivin Jan 9 '14 at 0:50
• 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? – User1234321232 Jan 9 '14 at 1:39

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.