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If $A$ is any $m\times n$ matrix then $AA^H$ and $A^HA$ are respectively $m\times m$ and $n \times n$ matrix. Then I observed that both the matrix share the same eigenvalue and if anyone has any other eigenvalue, that is $0$. I could not figure out the reason. Thanks in advance for any help.

Arindam
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    This is true more generally for $AB$ and $BA$ for any $m \times n$ and $n \times m$ matrices. See, for example, https://qchu.wordpress.com/2012/06/05/ab-ba-and-the-spectrum/. – Qiaochu Yuan Oct 25 '17 at 17:42
  • If $AB v = \lambda v$ with $\lambda \neq 0$ then $BA (Bv) = \lambda (Bv)$ (and $Bv \neq 0$). Hence if $\lambda $ is an eigenvalue of $AB$ it is an eigenvalue of $BA$. – copper.hat Oct 25 '17 at 17:53

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