Let $\mathbb{C}_{-}:=\{z\in\mathbb{C}:\operatorname{Re}(z)\le 0\}$. Let a one-step method have stability function $R:U\subset\mathbb{C}\to\mathbb{C}$, and suppose the method is A-stable, meaning that $\mathbb{C}_{-}\subset U$ and $|R(z)|\le 1$ for every $z\in\mathbb{C}_{-}$. Assume moreover that, when the method is applied to diagonal linear systems, it decouples componentwise with the same scalar stability function on each component.

Let $m\in\mathbb{N}$, let $A\in\mathbb{C}^{m\times m}$ be diagonal with diagonal entries $\lambda_1,\dots,\lambda_m\in\mathbb{C}_{-}$, and let $k>0$. If the method is applied with step size $k$ to

\begin{align*} y'(t)=Ay(t) \end{align*}

and produces numerical states $Y_n=(Y_{n,1},\dots,Y_{n,m})\in\mathbb{C}^m$, then $k\lambda_j\in U$ and

\begin{align*} Y_{n+1,j}=R(k\lambda_j)Y_{n,j} \end{align*}

for every $n\in\mathbb{N}\cup\{0\}$ and every $1\le j\le m$. Consequently,

\begin{align*} |Y_{n,j}|\le |Y_{0,j}| \end{align*}

for every $n\in\mathbb{N}\cup\{0\}$ and every $1\le j\le m$.