Suppose that $T, S \in L(V )$.
Find necessary and sufficient conditions for there to exist bases $\alpha , \alpha ' , \beta, \beta ' $ such that $ [T]_{\alpha}^{\beta} =[S]_{\alpha '}^{\beta '}$
i would like to make one of these matrices a diagonal matrices w.r.t the new basis's which i know how to do but im not sure how to also adjust the second one so it is exactly the same with the new basis...
It is necessary and sufficient that $S$ and $T$ have the same rank.
If $T$ has rank $k$, you may find bases $\alpha,\beta$ such that $[T]_\alpha^\beta$ has the canonical form $[T]_\alpha^\beta = I_k \oplus 0$. Indeed, take a basis $(e_i)$ for $\operatorname{im}(T)$ and complete it to a basis $\beta$ for $V$. Then pick $k$ linearly independent preimage vectors $(T^{-1}(e_i))$ and complete them to a basis $\alpha$ for $V$ using basis vectors for $\ker(T)$. Hence, if $S$ and $T$ have the same rank, you may find pairs of bases for both such that both have the same canonical form.
The converse follows from the well-known result that the rank of a linear map coincides with the rank of any of its matrix representations.