I am self studying working through the book "A First Course in the Numerical Analysis of Differential Equations" and have come to a deadend on q 1.2.
The linear system $y' = Ay, y(0) = y_0$, where $A$ is a symmetric matrix, is solved by the Euler method.
Let $e_n = y_n - y(nh)$
Where $y_n$ denotes the Euler approximation and $y(nh)$ the exact solution ($h$ is Euler step size).
Prove that $ ||e_n ||_2 = ||y_0||_2 max|(1+h\lambda)^n - e^{nh\lambda}|$
Where $\lambda \in \sigma(A)$ where $\sigma(A)$ is the set of eigenvalues of A.
I have tried various approaches such as writing $e_n$ as the error bound of the Euler method and taking the norm, but I can't seem to get $||y_0||_2$ in my answers.
Given that A is symmetric it can be diagonalised. Therefore we can write down the solution of the system of ODEs directly, and simply subtract the discreteized solution front the exact.