Consider that I have N different matrices $A_1,A_2,....A_N$ over finite fields, and all are non-singular. There exists a non-zero vector $x$ such that
$A_ix = A_jx$
for any (i,j) pair in the matrices.
Can you prove (or give a counter example) for the following claim
$x$ is an eigenvector for all $A_i$ for the eigen value 1.
Edit 1: I am looking at the case where the linear span of $A_i$ are not the entire of $k^{d\times d}$
Given any two vectors $x,y$ such that $x\ne 0$, the set of matrices $U(x,y)=\{A\in k^{d\times d}\,:\, Ax=y\}$ is an affine subspace of dimension $d^2-d$. What you are asking is whether there are some $x,y$ such that $\{A_1,\cdots, A_N\}\subseteq U(x,y)$. It is thus apparent that this cannot be the case if $\dim\operatorname{Span}(A_2-A_1,\cdots, A_N-A_1)>d^2-d$.