Let $A_1$, $A_2$, and $A_3$ be three different mutually orthogonal random hermitian operators, say dimension of $30\times 30$. For three random real numbers $a_1$, $a_2$, $a_3$, we have $$ (a_1A_1+a_2A_2+a_3A_3)\vec{v}=s\vec{v}. $$ i.e., $\vec{v}$ is an eigenvectors of the matrix $a_1A_1+a_2A_2+a_3A_3$ with eigenvalue $s$. We ask if there exists another set of real numbers $(a_1',a_2',a_3',s')$ such that $$ (a_1'A_1+a_2'A_2+a_3'A_3)\vec{v}=s'\vec{v}. $$ generically.

I have numerically verified that the $(a_1,a_2,a_3,s)$ is unique generically. Now in a more general setting, the number of random hermitian operators of size $d\times d$ is $r$. We ask if $(a_i,s)$ is unique for certain $\vec{v}$ when $r=O(d)$. Is there any mathematical theorem which guarantte this?