Hessian of a Quadratic Form

Start from the gradient $\nabla f(x)=2Ax$ and differentiate once more.

The Hessian is the Jacobian of the gradient.

1. 1. Differentiate the gradient

   Since $\nabla f(x)=2Ax$ is linear in $x$, its Jacobian is simply the coefficient matrix.

2. 2. Conclude

   Differentiating $2Ax$ with respect to $x$ gives the Hessian directly.

   $${\nabla }^{2}f(x)=2A$$

Answer: 2A