I have a multivariate function in one of whose critical points the Hessian matrix is singular. Is there any general method to determine the type of this critical point? Would be worth plotting the function in the direction of the eigenvector associated to a zero eigenvalue of the Hessian matrix?
Suppose that there is a function $f\left( \mathbf{x}_0 + \mathbf{\xi } \right) \approx f\left( {{\mathbf{x}}_{0}} \right)+\frac{1}{2}{{\mathbf{\xi }}^{\text{T}}}\mathbf{H}\mathbf{\xi }$, where $\mathbf{x}_0$ is a given stationary point, $\mathbf{\xi}=\mathbf{x}-\mathbf{x}_0$ is the vector of relative coordinates, and $\mathbf{H}=\mathbf{H}\left( {{\mathbf{x}}_{0}} \right)$ is the Hessian matrix at $\mathbf{x}=\mathbf{x}_0$. Then we can obtain the following form: $g\left( \mathbf{\zeta } \right)=2\left[ f\left( {{\mathbf{z}}_{0}}+\mathbf{\zeta } \right)-f\left( {{\mathbf{z}}_{0}} \right) \right]={{\mathbf{\zeta }}^{\text{T}}}\text{diag}\left( {{\lambda }_{i}} \right)\mathbf{\zeta }$, where ${{\mathbf{z}}_{0}}={{\mathbf{V}}^{\text{T}}}{{\mathbf{x}}_{0}}$ and $\mathbf{\zeta }={{\mathbf{V}}^{\text{T}}}\mathbf{\xi }$, $\lambda_i$ is the $i$th eigenvalue of H, and V is the eigenvector matrix of H. What should I plot to see the behavior of the function around $\mathbf{x}_0$?