Let $A(x_1,…,x_n)$ be an $n\times n$ matrix field over $R^n$.
I am interested in the partial derivative determinant of $A$ in respect to $x_i$. In can be shown that:
$\frac{\partial{\det(A)}}{\partial{x_i}} = \det(A)\cdot\sum_{a=1}^{n}{\sum_{b=1}^{n}{ A^{-1}_{a,b} \cdot \frac{ \partial{A_{b,a}} }{ \partial{x_i} }}}$
I was able to prove this using induction and careful, boring calculations, but I was wondering if there was any intuition behind this formula?
Best Answer
your identity follows simply by using $\log({\rm det}\; A)= {\rm tr}\; (\log A)$, so
$$\frac{\partial}{\partial x_i}{\rm det}\;A= \frac{\partial}{\partial x_i} \exp({\rm tr}\;\log A)= ({\rm det}\;A) \frac{\partial}{\partial x_i}{\rm tr}\;\log A= ({\rm det}\;A)\;{\rm tr}\;\left(A^{-1}\frac{\partial}{\partial x_i}A\right)$$
this identity is known as Jacobi's formula.