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?


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.