# Bound on the max norm of a matrix based on its Jordan canonical form

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?