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..

the same basis  is also a basis of $A+B$, because if $v$ is an element of this basis we have $(A+B)v=Av+Bv=\lambda v+\gamma v=(\lambda +\gamma)v$, so $v$ is an eigenvectors of $A+B$.
