[Math] How to compute the Hessian Matrix

Question

I want to compute the Hessian matrix of a 2-dimensional vector function.

\begin{pmatrix}x_1 + x_2 + x_3\\x_2^2 -x_1x_2\end{pmatrix}

Can anyone pleae explain how to compute this since I can find it nowhere..

Answer 1

The hessian matrix of a vector function is a $(1,2)$ tensor whose entries are given by $\partial_{i}\partial_{j}f^{k}$. Now i'll let you do the rest, Sjoerd, my friend.

Think of this as two matrices, each one the hessian of a scalar function $f^{1}$ or $f^{2}$. How would you compute the hessian of $f^{1}$? well $f^{1}$ is a first order function, so all the second derivatives are done for, $\partial_{i}\partial_{j}f^{1}=0$, now onto $f^{2}$ the only nonvanishing entries are $\partial_{2}\partial_{2}f^{2}$=$\partial_{2}\partial_{2}x_{2}^{2}$=2 and the mixed $\partial_{2}\partial_{1}f^{2}$=1. Of course the hessian is symmetric, so $\partial_{1}\partial_{2}f^{2}$=1. These are all the nonvanishing entries; just 3.