This question is a generalisation of Eigenvalues of $AB$ and $BA$ where $A$ and $B$ are square matrices.

Let $A$ and $B$ be $m\times n$ and $n\times m$ complex matrices, respectively, with $m < n$. If the eigenvalues of $AB$ are $\lambda_1, \ldots, \lambda_m$, what are the eigenvalues of $BA$?

If the matrices were square, then the conclusion would follow from the fact that $AB$ and $BA$ have the same characteristic polynomial. With rectangular matrices this is not going to happen; how to proceed then?

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    $\begingroup$ What is your attempt? $\endgroup$ – apnorton May 24 '14 at 17:29

Let $\lambda\neq 0$ be an eigenvalue of $AB$

Then, for some non-zero $v$, $ABv=\lambda v$

Hence $BABv=\lambda Bv$

Equivalently $(BA)(Bv)=\lambda (Bv)$

Note that $Bv \neq 0$. Otherwise, $ABv=\lambda v=0$, hence $\lambda=0$

Hence $\lambda$ is a non-zero eigenvalue of $BA$

Switching $A$ and $B$ in the previous proof, it also holds that a non-zero eigenvalue of $BA$ is a non-zero eigenvalue of $AB$

Conclusion: $AB$ and $BA$ have the same non-zero eigenvalues.

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  • 3
    $\begingroup$ You are not assuming that $\lambda \neq 0$? $\endgroup$ – zoli May 24 '14 at 18:01
  • $\begingroup$ Assumption that $\lambda \ne 0$ is crucial. Otherwise, it may happen that $Bv = 0$, which cannot be an eigenvector of $BA$. $\endgroup$ – Vedran Šego May 24 '14 at 18:11
  • $\begingroup$ @VedranŠego I thought the $\lambda_i$ were non-zero. Thanks for noticing :) $\endgroup$ – Gabriel Romon May 24 '14 at 18:25

This is a totally different approach, but way more powerful.

I'm going to prove that $\chi_{BA}=(-X)^{n-m}\chi_{AB}$ by elementary means.

Let $r=\operatorname{rank}(A)$

From a well-known theorem, derive that there exists $P,Q$ invertible $m\times m$ and $n \times n$ matrices such that $$A=P\begin{bmatrix}I_r& 0\\ 0 &0\end{bmatrix}Q $$

where $I_r$ denotes the $r\times r$ identity matrix.

By changes of basis, $$B=Q^{-1}\begin{bmatrix}E& F\\ G &H\end{bmatrix}P^{-1}$$

For some submatrices $E,F,G,H$.

Note that $AB=P\begin{bmatrix}E& F\\ 0&0\end{bmatrix}P^{-1}$ and $BA=Q^{-1}\begin{bmatrix}E& 0\\ G&0\end{bmatrix}Q$.

Hence $\chi_{AB}=\det(E-XI_r)(-X)^{m-r}$ and $\chi_{BA}=\det(E-XI_r)(-X)^{n-r}$

Hence $\chi_{BA}=(-X)^{n-m}\chi_{AB}$.

The results in the two other answers are now a simple consequence of the formula.

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Also, note that $BA$ will have an eigenvalue of $0$, since it's $n\times n$, but the maximum rank of each of $A$ and $b$ is $m<n$.

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the characteristic polynomials of $AB$ and $BA$ are still $\lambda^{m-r}p(\lambda)$ and $\lambda^{n-r}p(\lambda), p(0) \neq 0$ the reason is $tr(AB)^k = tr(BA)^k$ for all $k$. showing the coefficients of the characteristic polynomials are the same.

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  • $\begingroup$ This is a cool remark, but compared to DoDom's proof above, it does not work it positive characteristic (eg over finite fields). $\endgroup$ – ACL Apr 12 '15 at 16:33

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