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Here is a lemma that is part of Threorem $142$C in J. C. Butcher's book on numerical methods of ODEs for which I do not understand the proof. The lemma asserts that a) $\Rightarrow $ b), where A is an $m \times m$ matrix and a) and b) are the following:

a) The Jordan canonical form of $A$ has all its eigenvalues in the closed unit disc with all eigenvalues of magnitude $1$ lying in $1 × 1$ blocks.

b) There exists a non-singular matrix $S$ such that $\left \| S^{-1} A S \right \|_\infty \leq 1$.

And the proof (literally transcribed), reads

If a) is true, then $S$ can be chosen to form $J$, the Jordan canonical form of $A$, with the off-diagonal elements chosen sufficiently small so that $\left \| J \right \|_\infty \leq 1$.

How can we 'choose' the off-diagonal elements? Am I confusing the Jordan Normal form with something else?

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