Given $r$ complex matrices $A_1,\ldots,A_r$, each of size $m$-by-$n$, we say that a nonzero $x \in \mathbb{C}^n$ is a generalized eigenvector of $(A_1,\ldots,A_r)$ if
$$A_1x \wedge \cdots \wedge A_rx = \mathbf{0},$$
that is to say, if there is a non-trivial linear relation between $A_1x,\ldots,A_rx$.
One recovers the usual notion of eigenvector if $r=2$, $m=n$, $A_1 = A$ ($A$ is a complex $n$-by-$n$ matrix) and $A_2 = I$.
I have seen this generalization if $r=2$, and $m=n$. Has it been studied if $r>2$ by any chance? If so, does anyone have some references please?
If $V=\mathbb{C}^n$ and $W=\mathbb{C}^m$, then an $r$-tuple of $m$-by-$n$ matrices $(A_1,\ldots,A_r)$ defines an $r$-tuple $(s_1,\ldots,s_r)$, where
$$s_i \in H^0(\mathbb{P}(V), \mathcal{O}(1) \otimes W),$$
($1 \leq i \leq n$) corresponds to $A_i$. Thus for a generic $(A_1,\ldots,A_r)$, the algebraic set
$$s_1 \wedge \cdots \wedge s_r = \mathbb{0}$$
is a representative of $H_{2r}(\mathbb{P}(V),\mathbb{C})$ which is dual to
$$c_{r}(\mathcal{O}(1) \otimes W, \mathbb{C}) \in H^{2r}(\mathbb{P}(V),\mathbb{C})$$
via Poincare duality. I owe K. K-M. for the remark on the link with Chern classes.