Let's consider the Gell-Mann matrices $\vec{\lambda} =(\lambda_1, \lambda,_2,\cdots, \lambda_8)$. Another hermitian matrix can be constructed in terms of a real vector $\vec{a}=(a_1,\cdots,a_8)$ by $A = \vec{a} \cdot \vec{\lambda}$. Is there a neat relationship between the eigenvalues $E_i$, the eigenvectors $v_i$ of $A$ and the vector $\vec{a}$? More clearly, is there a closed expression $A \vec{v}_i(\vec{a}) = E_i(\vec{a})\vec{v}_i(\vec{a})$? I tried to use the implicit function theorem to derive something, but I end up with not very well defined differential equation which I cannot solve. Any ideas?
PS: I am not a mathematican, please feel free to add/ remove/ change the text and or the tags.