eigen_II#

  • Series: lecture-python-intro

  • File: lectures/eigen_II.md

  • Audit date: 2026-08-26

  • Corpus snapshot: a12d17c0ef

  • Categories audited: writing, math, code, references, links, admonitions (JAX out of scope)

  • Overall score: 8.4 / 10

  • Priority: LOW

Score breakdown#

Category

Score

One-line note

Writing

4.5/10

qe-writing-004 ×2; qe-writing-005 ×3; qe-writing-003 ×2, +4 more.

Math

9.5/10

qe-math-009 ×1.

Code

7.5/10

qe-code-001 ×7.

JAX

out of scope

JAX rules target lecture-jax.

Figures

N/A

no figures or plotting code.

References

9/10

qe-ref-001 ×1.

Links

10/10

no mechanical violations detected.

Admonitions

10/10

no mechanical violations detected.

Issues#

Critical#

None found.

High severity#

  • [qe-code-001] (reviewer) — Follow PEP8 unless closer to mathematical notation. Count: 7. Lines: 138, 139, 148, 149, 322, 550, 561. Example: {v[:,0]}, {-v[:,1]}, {w[:,0]}, {-w[:,1]} at 138-139 and {e[:,0]} … {ε[:,1]} at 148-149 omit the space after the comma in the index, which the same lecture writes correctly as v[:, i] at 311-312; def check_convergence(M) at 322 follows the previous top-level def with a single blank line (PEP8 asks for two); #dominant eigenvalue/spectral radius at 550 has no space after the hash; d.shape = (3,1) at 561 has no space after the comma.

Medium severity#

  • [qe-writing-001] — Use one sentence per paragraph. Count: 1. Lines: 267. Example: 2 sentences in one paragraph.

  • [qe-writing-002] (reviewer) — Keep writing clear, concise, and valuable. Count: 2. Lines: 63, 222. Example: line 63 packs the irreducibility definition into 33 words and then mis-states its own condition: “\(A + A^2 + A^3 + \cdots \gg 0\), where \(\gg 0\) indicates that every element in \(A\) is strictly positive” - the positivity is a property of the sum, not of \(A\); line 222 “We know that in real world situations it’s hard for a matrix to be everywhere positive (although they have nice properties)” leaves “they” and the parenthetical unattached to anything.

  • [qe-writing-003] (reviewer) — Maintain logical flow. Count: 2. Lines: 191, 453. Example: line 191 says “Let’s build our intuition for the theorem using a simple example we have seen before” but the next section (195-203) introduces a fresh matrix and never returns to mc_eg1, so the promised link is not made; line 453 “Recall that eigenvalues are ordered from smallest to largest from \(i = 1 ... n\)” asks the reader to recall a convention this lecture has not stated - and the code at 305-308 picks the dominant eigenvalue with np.max/np.argmax, giving no ordering at all.

  • [qe-writing-004] — Avoid unnecessary capitalization in narrative text. Count: 2. Lines: 457, 554. Example: mid-sentence ‘Theorem’.

  • [qe-writing-005] (reviewer) — Use bold for definitions, italic for emphasis. Count: 3. Lines: 63, 118, 265. Example: definitions left unbolded while emphasis is bolded - the exact reversal the rule forbids: “is called irreducible” at 63 and “is called a left eigenvector” at 118 carry no bold, although nonnegative (49), dominant eigenvalue (164), primitive (228) and Perron projection (267) are all correctly bolded; at 265 bold is used for emphasis instead, “the inequality \(|\lambda| \leq r(A)\) is strict”.

  • [qe-writing-007] (reviewer) — Use visual elements to enhance understanding. Count: 3. Lines: 169, 355, 465. Example: the lecture has no figure at all across 569 lines, and three passages are about things only a picture shows: the Perron-Frobenius statement at 169-184 compares \(|\lambda|\) against \(r(A)\) for every eigenvalue, which is a plot of the spectrum against a circle of radius \(r(A)\); the convergence of \(r(A)^{-m} A^m\) to the Perron projection is delivered at 355-362 as printed Frobenius norms for n = 1, 10, 100, 1000, 10000 rather than an error curve; and the \(O(\eta^t)\) rate at 462-468 is asserted with no plot of \(\|\psi P^t - \psi^*\|\) against the spectral gap.

Low severity#

  • [qe-math-009] (reviewer) — Choose simplicity in mathematical notation. Count: 1. Lines: 447. Example: \(\psi \in \mathscr{D}(S)\) at 447 reaches for a script alphabet for the set of distributions on \(S\) - the only decorative symbol in a lecture that otherwise uses plain letters - and the symbol is never defined here or linked to where it is defined.

  • [qe-ref-001] — Use correct citation style. Count: 1. Lines: 443. Example: {cite} in narrative flow: ‘in {cite}’.

  • [qe-writing-008] — Remove excessive whitespace between words. Count: 1. Lines: 266. Example: 2 spaces.

Strengths#

  • The lecture is stitched into the series rather than standing alone - {ref} and {doc}-style links reach mc_irreducible (57), la_eigenvalues (104), mc_eg1 (191), mc1_ex_1 (386, 431), mc_eg3 and mc_eg2 (395), and la_neumann (534) - so every borrowed concept has a destination.

  • Transpose is ^\top in all six places it appears (118, 120, 156, 267, 440, 450); no apostrophes and no ^T.

  • The theorem is split into two prf:Theorem blocks with continuing claim numbers - 1-5 at 169-184 and 6-7 at 260-268 - and each numbered claim is then checked back one by one in the worked lists at 213-217 and 290-295.

  • The non-primitive counterexample at 369-380 runs check_convergence on a matrix that fails, immediately after the primitive cases, so the necessity of primitivity is demonstrated instead of asserted.

  • Greek names in code are Unicode throughout - λ (129, 132, 550), ε (145, 149), ψ_star (407, 421).