Can someone please expand on this idea from wikipedia https://en.wikipedia.org/wiki/Jordan_normal_form, under Generalized Eigenvector - Uniqueness.

Particularly, how you can use the rank of $(A - \lambda I)^{k_1-2}$ to find the the number of Jordan Blocks for $k_1-1$. But then it says the general case is similar, but I do not quite understand what the next step given further ranks. My question is how would you use the rank of $(A - \lambda I)^{k_1-3}$ to determine the number of Jordan blocks for $k_1-2$ and so on?

  • $\begingroup$ Think about what happens to the nullity of successive power of $A-\lambda I$ in terms of the Jordan chains for each independent eigenvector. $\endgroup$ – amd Apr 21 at 20:09

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