I'm working out of Deuflhard and Bornemann's Scientific Computing with ODEs (Springer 2000).

The statement is:

Moreover, $\tau_c = \tau_c^+$ holds if all eigenvalues $\lambda \in \sigma(A)$ ... have index $\iota(\lambda)=1$.

In my case, I've found that $A$ has two eigenvalues, $\lambda = \{1,1000\}$. Being more of a programmer, I tend to think of the index of the position in the set, i.e. index(1) = 0, index(1000) = 1, etc. -- but its pretty clear from the rest of the section that there is some other definition of index that I'm not aware of.

I can do most of the leg work myself learning about it, but googling permutations of "eigenvalue index" haven't been very helpful.

Thanks in advance.

  • 2
    $\begingroup$ The definition here seems very plausible, but you've probably seen that already. $\endgroup$ – Dylan Moreland Dec 15 '11 at 3:03

This says:

The index $\iota(\lambda)$ of an eigenvalue $\lambda\in\sigma(A)$ is the maximal dimension of the Jordan blocks of $A$ containing $\lambda$.

$\sigma(A)$ here is the set of eigenvalues (the spectrum) of $A$. Thus, the condition $\iota(\lambda)=1$ holding for all the eigenvalues means that the matrix is diagonalizable.

  • $\begingroup$ Thanks! This is it and I seem to have just overlooked the definition a hundred or so pages earlier. $\endgroup$ – jedwards Dec 15 '11 at 3:14

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