Hessian and quadratic form

Question

I can't understand what my professor meant in our Calc III book. In a demostration, he wrote something like "we are going to refer to the hessian ($Hf_{P}$) and its associated quadratic form ($Q_{P}(V)=\frac{1}{2}\langle Hf_{P}V,V \rangle$) indistinctly.". As far as I know, in a quadratic form ($Q(X)=\langle XA,X \rangle$), $A=a_{ii}$ is given by $$a_{ij} = \begin{cases} a_{ii}, & \text{if } i=j,\\ \frac{a_{ij}+a_{ji}}{2}, & \text{if } i\neq j. \end{cases}$$ Then, how is it possible to refer to them indistinctly?

Answer 1

Given the quadratic form $Q,$ and column vector $X,$ the matrix $H$ is $$ h_{ij} = \frac{\partial^2 Q}{\partial x_i \partial x_j} $$ and the form is then $$ Q(X) = \frac{1}{2} X^T H X $$

For example, take form $$ x^2 + 2 y^2 + 3 z^2 + 4 yz + 5 zx + 6xy \; , \; $$ the Hessian matrix is $$ H = \left( \begin{array}{ccc} 2 & 6 & 5\\ 6 & 4 & 4 \\ 5 & 4 & 6 \end{array} \right) $$