Computing Laplacian of $ (Bx)\cdot x$

Question

Could you help me and tell me how I should find the $\Delta((Bx) \cdot x)$ if $B \in \mathbb{R}^{N\times N}$ and $x \in \mathbb{R}^N$ ?

All I can think of is writing $\Delta((Bx) \cdot x)= \operatorname{div}(D(Bx \cdot x))$. The final answer is $2\operatorname{trace}(B)$, but I have no idea how I should get to trace of $B$. Please help me what way I should use.

Answer 1

In standard Euclidean coordinates, the Laplacian is the trace of the Hessian: $$\Delta f = \operatorname{tr}(Hf) = \sum_i \frac{\partial^2 f}{\partial x_i^2}.$$

Now:

1. Compute the Hessian of $f(x) = Bx \cdot x.$ Show that the Jacobian is $x^T(B+B^T)$ and the Hessian is $B+B^T$.
2. Use the properties of trace (in particular, linearity and invariance under transpose) to get the formula $\Delta f = 2\operatorname{tr}(B)$.