where is diagonalizable; , that the solution given by forward Euler is
where we consider the Forward Euler method to be N
Now, I want to show that if the real parts of (the eigenvalues of and thus the entries of ) are negative, then for all implies that .
Now, in order to show this, based on what we have obtained for the formula for , we need only show that for each . I am having a really tough time doing this, and have mostly just tried to play around with AM-GM stuff.