This is a homework problem I have a hard time to understand. Any tips would be appreciated to get me in the right direction.
Given functions $f: \mathbb{R}^n \to \mathbb{R}$ and $\boldsymbol{g}: \mathbb{R}^n \to \mathbb{R}^m$ you can minimize $f(\boldsymbol{x})$ given the constraint $\boldsymbol{g}(\boldsymbol{x}) = \boldsymbol{0}$ by solving the gradiant of the Lagrangian function to zero: $ \bigtriangledown \mathcal{L}(\boldsymbol{x}, \boldsymbol{y}) = \begin{bmatrix} \bigtriangledown f(\boldsymbol{x}) + \boldsymbol{J}_g^T(\boldsymbol{x})\boldsymbol{\lambda} \\ \boldsymbol{g}(\boldsymbol{x}) \end{bmatrix} = \boldsymbol{0} $ Where $\boldsymbol{J}_g^T(\boldsymbol{x})$ is the Jacobian matrix of $\boldsymbol{g}(\boldsymbol{x})$.
The Hessian of this matrix can be computed as follows. $ \boldsymbol{H}_{\mathcal{L}} (\boldsymbol{x}, \boldsymbol{y}) = \begin{bmatrix} \boldsymbol{B}(\boldsymbol{x}, \boldsymbol{y}) & \boldsymbol{J}_g^T(\boldsymbol{x}) \\ \boldsymbol{J}_g(\boldsymbol{x}) & \boldsymbol{0} \end{bmatrix} $ Where $\boldsymbol{B}(\boldsymbol{x}, \boldsymbol{y}) = \boldsymbol{H}_f(\boldsymbol{x}) + \sum_{i=1}^m \lambda_i \boldsymbol{H}_{gi}(\boldsymbol{x})$
How can I prove that $\boldsymbol{H}_{\mathcal{L}} (\boldsymbol{x}, \boldsymbol{y})$ can not be positive definite?