A lemma in Gerschgorin theorem

Let :

  • $n geq 2$
  • $A in mathcal{M}_n( mathbb{C} )$
  • $forall i leq n , |a_{i,i}| > sum_{j ne i}^{n} |a_{i,j}| $
  • $V in mathcal{M}_{n,1}( mathbb{C} ), V ne 0 , Y=AV$
  • $V=(v_i)_i$ and $Y=(y_i)_i$

We want to prove that :
$$forall i leq n , |a_{i,i}| |v_i| – sum_{j ne i}^{n} |a_{i,j}| |v_j| leq |y_i|$$


My attempt :
$sum_{j ne k} a_{i,j} z_j + a_{i,k} z_k =0$