Jacobian of vec of a matrix into vec of its inverse

Question

Suppose that $X$ is invertible and $n\times n$ and the transformation: $$ \varphi:\operatorname{vec}(X)\mapsto\operatorname{vec}(X^{-1}). $$ For example, with $n=2$ and $X=\begin{pmatrix}a & b \\ c &d\end{pmatrix}$, we have $$ \varphi:\begin{pmatrix}a \\ c \\ b \\ d\end{pmatrix}\mapsto\frac{1}{ad-bc}\begin{pmatrix}d \\ -c \\ -b \\ a\end{pmatrix}. $$ Can you point me to some references (a textbook is my most preferred source for this context) that deal with Jacobian of $\varphi$.

Answer 1

I haven't any book at hand, but the usual argument shows that the derivative of $X\mapsto X^{-1}$ is the linear map $H\mapsto-X^{-1}HX^{-1}$, meaning that the Jacobian matrix of $\varphi$ is $J=-(X^{-1})^T\otimes X^{-1}$. So, in your example, we should have $$ \pmatrix{ \dfrac{d\varphi_1}{da}&\dfrac{d\varphi_1}{dc}&\dfrac{d\varphi_1}{db}&\dfrac{d\varphi_1}{dd}\\ \dfrac{d\varphi_2}{da}&\dfrac{d\varphi_2}{dc}&\dfrac{d\varphi_2}{db}&\dfrac{d\varphi_2}{dd}\\ \dfrac{d\varphi_3}{da}&\dfrac{d\varphi_3}{dc}&\dfrac{d\varphi_3}{db}&\dfrac{d\varphi_3}{dd}\\ \dfrac{d\varphi_4}{da}&\dfrac{d\varphi_4}{dc}&\dfrac{d\varphi_4}{db}&\dfrac{d\varphi_4}{dd}} =\frac1{\det(X)^2}\pmatrix{-d^2&db&cd&-cb\\ dc&-da&-c^2&ca\\ bd&-b^2&-ad&ab\\ -bc&ba&ac&-a^2}. $$