Suppose $A$ is a $n \times m$ matrix and $B$ is a $m \times n$ matrix. Then it is known that $det(I\_{n}+AB)=det(I\_{m}+BA)$.


Is there an analogous identity of the form $det(P\_{1}+AB)=det(P\_{2}+BA)$, where $P\_{1},P\_{2}$ are positive definite? Or something like it?



determinantal identity sought
Given $P$, $\det(P+AB)$ does not depend only on $BA$. For example, take $n=m=2$,
$P = \pmatrix{2 & 0\cr 0 & 1\cr}$, $A = \pmatrix{1 & t\cr 0 & 1\cr}$, $B = \pmatrix{0 & 1\cr
1 & -t\cr}$. Then $\det(P + AB) = 1 - t$ depends on $t$, but $BA = \pmatrix{0 & 1\cr 1 & 0\cr}$ doesn't depend on $t$. So any $P\_2$ such that $\det(P+AB) = \det(P\_2+BA)$ must depend on $t$.


