Hessian Matrix of $(Ax-b)$

Question

I'm confused about the way of finding the Hessian matrix of simple matrix $(Ax-b)$.

After learning a few explanations, I figured out $(Ax-b)$ can be written as $(Ax-b)^t(Ax-b)$ and the gradient/derivative is $2A^t(Ax-b)$ but I'm still confused on finding the second derivative for it.

Answer 1

You've successfully calculate the gradient of the function as $$g=2A^T(Ax-b)$$ Now you want the Hessian. The calculation is straightforward. $$\eqalign{ dg &= 2A^T(A\,dx-0) = 2A^TA\,dx \\ \frac{\partial g}{\partial x} &= 2A^TA = H \qquad\big\{{\rm Hessian}\big\} \\ }$$

Answer 2

The hessian of $x^TMx$ is $M+M^T$ ($=2M$ if M is symmetric) for a quadratic matrix $M$ and the hessian of $x^Tv$ is $0$ for every vector $v$.