The linear system ${y}^{\prime}=Ay,y(0)={y}_{0}$, 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.

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.