Given $\mathbf{J}$, the Jacobian matrix of a function $\mathbf{F}:\mathbb{R}^n\to\mathbb{R}^n$, the system $$ \mathbf{J}\mathbf{v}=\lambda \mathbf{v} $$ has solutions corresponding to eigen-pairs $(\mathbf{v},\lambda)$, where $\mathbf{v}$ is an eigenvector of $\mathbf{J}$ and $\lambda$ its corresponding eigenvalue.
Is there a generalisation of the definition of eigenvector and eigenvalue involving the Hessian $\mathbf{H}$ of a function $F:\mathbb{R}^n\to \mathbb{R}$? That is, is there a name for $(\mathbf{v},\lambda)$ satisfying $$ \mathbf{H}\mathbf{v}=\lambda \mathbf{v} $$ Alternatively (and perhaps more relevant), when $n=2$ (for simplicity) and considering the vector-valued Taylor expansion, is there any known eigen-based generalisation of $(\mathbf{v},\lambda)$ satisfying $$ \frac12 \begin{pmatrix} \mathbf{v}^T \mathbf{H}_1(\mathbf{x})\mathbf{v}\\ \mathbf{v}^T \mathbf{H}_2(\mathbf{x})\mathbf{v} \end{pmatrix}= \lambda \mathbf{v} $$ where, in this case, $\mathbf{F}=(F_1,F_2)$ and $\mathbf{H}_{1,2}$ are the Hessian matrices of $F_{1,2}$, respectively.
Any ideas?
