A relationship between characteristic values and vectors

It is shown that for all nonzero $n$-component column vectors $\alpha$ and $\beta$ over a field $F$ there exists a set $\Gamma$ of $n$-square matrices over $F$ of cardinality ${n^2} - 2n + 2$ such that, for each $n$-square matrix $A$ over $F,A\alpha = \alpha$ or ${A^T}\beta = \beta$ if and only if 1 is a characteristic value of PA for every $P \in \Gamma$.