Before I state my question, I'll review the definition of $H$-eigenvalues / eigenvectors from L. Qi, Journal of Symbolic Computation, 2005. Suppose that I have a tensor $\mathbf A \in \mathbb R^{[k, n]}$, with entries $a_{i_1, i_2, \dots, i_k}$. Given a vector $x \in \mathbb R^n$, we write $\mathbf A x^{n - 1}$ to denote the vector in $\mathbb R^n$ with entries $$(\mathbf A x^{n - 1})_i = \sum_{i_2, \dots, i_k = 1}^n a_{i, i_2, \dots, i_k} x_{i_2} \cdots x_{i_k}$$ If a pair $\lambda \in \mathbb R$, $x \in \mathbb R^n$ satisfy $$\mathbf A x^{n - 1} = \lambda x^{[n - 1]}$$ where $x^{[n - 1]} \in \mathbb R^n$ has elements $(x^{[n - 1]})_i = x_i^{n - 1}$, we say that $\lambda, x$ are an $H$-eigenvalue / $H$-eigenvector of $\mathbf A$.
The $H$-eigenvector seems to generalize the right eigenvector of a matrix. My question: is there a similar generalization of the left eigenvector?