QuantEcon / QuantEcon/lecture-python-intro

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

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enhancement
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Jupyter Notebook
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65
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32
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4d 14h
Merged PRs (30d)
6

Description

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.

Contributor guide

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First steps

  1. Read the whole issue, then the project's contributing guide.
  2. Comment on the issue to say you are picking it up — it saves two people doing the same work.
  3. Fork the repository and make your change on a branch.
  4. Open a pull request that references the issue number.

Research direction

Start in eigen_I.md at the existing power-iteration discussion and read the surrounding eigenvalue material. Add the requested introductory treatment of algebraic and geometric multiplicity, a concrete 2×2 defective-matrix example, and a cross-reference explaining the precondition; leave Jordan normal form and state-space consequences to linear_models.md and issue #1018.

Written by the indexing model from the issue text.

Assessment

Tech stack
jupyter-notebook
Domain
documentation
Issue type
Documentation
Difficulty
2/5
Estimated time
1-3 hours
Activity status
Quiet
Clarity
Clearly specified
Newbie friendliness
78/100

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