Calculating the Hessian of a function $f :\mathbb{R}^{n} \rightarrow \mathbb{R}$ given an expression for the gradient including eigenvalues

Question

We are given a function $f :\mathbb{R}^{n} \rightarrow \mathbb{R}$ such that:

$\nabla{f}(x) = \lambda x$

where $\lambda$ is a scalar, and an eigenvalue of some Matrix $A \in M_{n \times n}$ that we do not know

$x \in \mathbb{R}^{n}$.

What is the Hessian?

I believe the solution is $\lambda I$, but want to confirm

Is there anything interesting about this result? Does it provide some interesting insight about the structure of $f$

Answer 1

The function $\nabla f$ is linear, hence its derivative at any point is just $\nabla f.$ That is to say, $Hf(x) = \nabla f = \lambda I.$ Therefore, $Hf(x) \cdot (h,k) = \lambda (h|k),$ where $(h|k)$ is the standard Euclidian product.