Is it possible to extract the phase of a determinant without computing the full determinant?
More explictly, given a complex matrix $U$, the determinant can be written in the form \begin{equation} \text{det}(U) = r e^{i\theta}. \end{equation} Is it possible to extract $\theta$ from $U$ without computing the full determinant?
Edit : As pointed by Hans Engler below, this can be done if we are happy with more work than computing the original determinant. I am looking for a method that is more efficient than computing the determinant itself.
Yes, it's possible. Here is a way that is unfortunately a lot more work. Compute $W = (U^\ast)^{-1} U$. Then $\det W = e^{2i \theta}$.
You have found $\theta$ without computing $r$.