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eigen_I: define diagonalisability and multiplicity — the power-iteration precondition is never explained #808

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@mmcky

eigen_I.md teaches eigenvalues, eigenvectors and power iteration over 33 KB, and uses the word "diagonalizable" exactly once — at the point where it introduces power iteration:

Power iteration is a method for finding the greatest absolute eigenvalue of a diagonalizable matrix.

The term is never defined, and the lecture contains no mention of algebraic or geometric multiplicity, defective matrices, or repeated eigenvalues. So the lecture states a precondition for a method it teaches, and gives the reader no way to know when that precondition holds or what happens when it fails.

Suggested scope

A short addition, sized to the intro series:

  • When diagonalisation fails — algebraic versus geometric multiplicity, and what makes a matrix defective
  • A worked example — the smallest defective matrix is 2×2 with a repeated eigenvalue and a one-dimensional eigenspace, which is concrete enough to show rather than assert
  • A sentence at the power-iteration precondition connecting back, so the one existing use of "diagonalizable" stops being unexplained

The Jordan normal form itself is not proposed here — that and its consequences for linear state-space transients stay in QuantEcon/lecture-python.myst#1018.

Why the split

Both halves descend from Tom Sargent's brief in QuantEcon/meta#28, which filed Jordan form under "Linear state-space dynamics". Reviewing that in QuantEcon/meta#344, the decision was to split it: the prerequisite belongs here, because a reader who meets a defective matrix in the intro series should not have to reach the intermediate series to learn what happened, and the gap stands on its own independently of state-space models. The dynamics consequence — how a Jordan block produces polynomial-times-exponential transients rather than pure exponentials — stays with linear_models.md in lecture-python.myst.

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