QuantEcon / QuantEcon/lecture-python-intro
eigen_I: define diagonalisability and multiplicity — the power-iteration precondition is never explained
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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
- Read the whole issue, then the project's contributing guide.
- Comment on the issue to say you are picking it up — it saves two people doing the same work.
- Fork the repository and make your change on a branch.
- 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