Suppose I have a $2N\times 2N$ matrix $X$ with the first half of the (real) eigenvalues $\lambda_k$ positive and the latter half negative. I want to compute $$P = f(\lambda_1)...f(\lambda_N)f(-\lambda_{N+1})...f(-\lambda_{2N})$$ if I didn't have the signs (or if $f$ was even etc), this would be $\det(f(X))$.
Is there any simple way to express this product, perhaps using permanents or Pfaffians?
Another way to state the above, which might help: suppose $X=S^{-1}\Lambda S$ with $\Lambda={\rm diag}(\lambda_1,\dots \lambda_{2N})$. Then what I am after is any way to express
$$ \det f(gSXS^{-1}) $$ where $$ g = {\rm diag}(\overbrace{1,1,\dots}^{N},\overbrace{-1,,\dots ,-1}^{N})$$ without making direct reference to $S$ - ie I'd like to not have to actually have diagonalised $X$ if at all possible.