Hessian after coordinate changing Let $f\colon \Bbb R^n\to\Bbb R$.
Let $z=Px$ coordinate changing. $P$ is $n\times n$ constant matrix, $x$ and $z$ are the variables in $\Bbb R^n$.
Does anyone know a formula which express how the Hessian of $f$ changed after this coordinate transform?
Thanks. 
 A: The chain rule is not valid for the second derivative except if the change of variable is linear, that is here the case.
Let $g(z)=f(P^{-1}z)$. The derivative is $Dg_z(H)=Df_{P^{-1}z}P^{-1}H$ and the second derivative is $D^2g_z(H,K)=D^2f_{P^{-1}z}(P^{-1}H,P^{-1}K)$ or $H^THess(g)_zK=H^TP^{-T}Hess(f)_{P^{-1}z}P^{-1}K$, that implies $Hess(g)_z=P^{-T}Hess(f)_{P^{-1}z}P^{-1}$.
EDIT. Answer to @jjjjjj. 
If $g(z)=f(\psi(z))$ where $\psi$ is a non-linear diffeomorphism, then, in a critical point, the result is similar; yet, the calculation is completely different.
$Dg_z(h)=Df_{\psi(z)}(D\psi_z(h))$ and 
$D^2g_z(h,k)=D^2f_{\psi(z)}(D\psi_z(h),D\psi_z(k))+Df_{\psi(z)}(D^2\psi_z(h,k))$.
If $z$ is critical, that is, $Df_{\psi(z)}=0$, then 
$D^2g_z(h,k)=D^2f_{\psi(z)}(D\psi_z(h),D\psi_z(k))$.
A: If I understand your question correctly, you are looking for a formula for hess(Ax) where A is a constant matrix.  It comes out to be A^T Hess(x) A.  You can use the chain rule to get there.  It is also helpful to realize that the hessian is the Jacobian of the gradient.
