$$\det(tI-AB)=t^{n-m} \det(tI-BA)$$
where $A$ and $B$ are $m \times n$ and $n \times m$ matrices, respectively, or is there any similar relations between the characteristic equations of transpose matrices.
Note. I think I have read somewhere that it holds when $B=A^T$...
The $t=1$ case of this result is Sylvester's determinant identity: for $m \times n$ and $n\times m$ matrices $A$ and $B$, $$ \det(I - AB) = \det(I - BA).$$
We can generalize to $tI$ without much trouble, since for a $k \times k$ matrix $M$ we have $\det(tM) = t^k \det(M)$: $$ \det(t I - AB) = t^m \det(I - (\tfrac1tA)B) = t^m \det(I - B(\tfrac1tA)) = t^{m-n} \det(t I - BA). $$
