1
$\begingroup$

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.

$\endgroup$
  • 2
    $\begingroup$ The definition here seems very plausible, but you've probably seen that already. $\endgroup$ – Dylan Moreland Dec 15 '11 at 3:03
6
$\begingroup$

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.

$\endgroup$
  • $\begingroup$ 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

Your Answer

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

Not the answer you're looking for? Browse other questions tagged or ask your own question.