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Let say $A \in \mathbb{F}^{n\times m}$ and $B \in \mathbb{F}^{m\times n}$. I am wondering weather there is any standard result stating the relationship between eigen values of $AB$ and $BA$ matrices.

I know that eigen values are same for $AB$ and $BA$ when both $A$ and $B$ are square matrices. But in a general setup, we can not use the result that- $\det(AB) = \det(BA)$.

Rajat
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    Do you understand the hint given here? A subtlety (partly hidden in that hint) is that $AB$ and $BA$ only share non-zero eigenvalues. – Jyrki Lahtonen Aug 11 '15 at 07:39
  • One is an $m\times m$ matrix and the other is an $n \times n$ matrix. One has $m$ eigenvalues while the other has $n$ eigenvalues. At least, from the trace, you can see that the sum of 2 sets of eigenvalues are the same. – mastrok Aug 11 '15 at 07:44
  • Actually, I was going through one pdf. They also mentioned the same thing, but no proof was given in it. Thanks for sharing the link. I will go through it. @JyrkiLahtonen – Rajat Aug 11 '15 at 07:46
  • Hi, @mastrok , yes that is obvious, but I think we can say something more stronger than that as it is mentioned in Jyrki 's comment. – Rajat Aug 11 '15 at 07:51
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    Look here or the second answer here. – el_tenedor Aug 11 '15 at 07:58

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