Is A+B digonalizable if they share the same basis of eigenvectors I was given the following statement:

I know that the sum of two diagonalizable matrices is not allways diagonalizable,
but i'm not sure how the added element of the shared base contributes..
I would very much appreciate an explanation or some guidance..

You can either see that, as matrices,
$A=PD_1P^{-1}, \quad B=PD_2P^{-1} \implies A+B=P(D_1+D_2)P^{-1},$
or simply verify that the eigenvectors of $A+B$ will be the same of $A$ and $B$, hence you'll have a basis of eigenvectors, and the linear map is thus diagonalizable.
