# Chain rule for Hessian matrix

Given $f\colon \mathbb{R}^n\rightarrow \mathbb{R}$ smooth and $\phi \in GL(n)$. What is the Hessian matrix $H_{f\circ \phi} = \left(\frac{\partial ^2 (f\circ \phi)}{\partial x_i\partial x_j}\right)_{ij}$?

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Denote $H_g(x)$ the Hessian matrix of a function $g$. Denote $g=f\circ \phi$. By the chain rule, we have $$D(f\circ\phi)(x)\cdot h=D(f(\phi x))\cdot D\phi\cdot h=D(f(\phi x))\cdot \phi\cdot h$$ hence $D(g)(x)=D(f(\phi x))\cdot \phi$. In particular, $$\partial_j g(x)=\sum_{k=1}^n\partial_kf(\phi x)a_{kj},$$ where $a_{kj}$ is the $(k,j)$-th entry of $\phi$.We can do the same, for a fixed $k$, for the map $x\mapsto \partial_kf(\phi x)$. We get \begin{align} \partial_{ij}f(\phi x)&=\sum_{k,l=1}^n(H_f(\phi x))_{lk}a_{li}a_{kj}\\ &=\sum_{k=1}^n(\phi^tH_f(\phi x))_{ik}a_{kj}\\ &=(\phi^tH_f(\phi x)\phi)_{ij}. \end{align}