C2_w3 Hessian(0,0) ? confused

in the video about concavity I do not understand he mentions H(0,0)=[[4 -1][-1 6]] How is H tied to this specific coordinate of (0,0)? It must have something to do with eigen value. Please explain

Hi @Hassan_Mohamed6!

The Hessian matrix is tied to a single value in the function’s domain. This is why we always talk about the hessian in a given point. So, for a given point in (x_0,y_0) \in \mathbb{R}^2, the Hessian at this point is given by

H(x_0,y_0) = \left[\begin{array}{cc} \dfrac{\partial f(x_0,y_0)}{\partial x^2} & \dfrac{\partial f(x_0,y_0)}{\partial x \partial y} \\ \dfrac{\partial f(x_0,y_0)}{\partial y \partial x} & \dfrac{\partial f(x_0,y_0)}{\partial y^2} \end{array} \right]

So, in that particular example in the lecture, Luis was looking into the concavity in the point (0,0).

The reason behind the Hessian changes in each point is because for each point in the function, the tangent plane at that point may change, therefore there will be another associate matrix.


Thanks for the explanation.