The identity for the determinant of $A$ in the title is well know in matrix analysis and comes from the Jacobi's formula. I am interested in the existence of nontrivial formulas like this one (they do not necessarily need to involve extensions of analytic functions to matrices but they should nevertheless not be trivial). In particular, I would like to know of identities of the form $\det(A)=f\circ g\circ h (A)$ where $g$, unlike in the case of the trace, verifies that $g(JABC)=g(JACB)$ whenever $J$ is a signature matrix (only $\pm1$ in the diagonal and the rest $0$) and $A$, $B$, $C$ are arbitrary real symmetric matrices (and even better if the matrix extension of the map $f$ and the map $h$ are inverse maps).
$\begingroup$
$\endgroup$
2
-
1$\begingroup$ What does "the matrix extension of the map $f$ and the map $h$" mean? $\endgroup$– LSpiceCommented Jul 18, 2022 at 17:43
-
1$\begingroup$ @LSpice An analytic map $f\colon\mathbb{C}\to\mathbb{C}$ can be extended through its Taylor expansion to a map $f\colon\mathcal{M}_{n}(\mathbb{C})\to\mathcal{M}_{n}(\mathbb{C})$ for any $n$. So I mean that this extension and the map $h\colon\mathcal{M}_{m}(\mathbb{C})\to\mathcal{M}_{m}(\mathbb{C})$ are inverse maps. But I also accept answers that do not verify this or that do not come from extensions of analytic functions to matrices since the main thing is the condition for $g$. $\endgroup$– HvjurthukCommented Jul 18, 2022 at 18:07
Add a comment
|