eigen_I#

Series: lecture-python-intro
File: lectures/eigen_I.md
Audit date: 2026-08-26
Corpus snapshot: a12d17c0ef
Categories audited: writing, math, code, figures, links, admonitions (JAX out of scope)
Overall score: 7.4 / 10
Priority: HIGH

Score breakdown#

Category	Score	One-line note
Writing	3.5/10	`qe-writing-004` ×5; `qe-writing-005` ×2; `qe-writing-003` ×2, +3 more.
Math	10/10	no mechanical violations detected.
Code	7.5/10	`qe-code-001` ×5.
JAX	out of scope	JAX rules target `lecture-jax`.
Figures	3.5/10	`qe-fig-003` ×7; `qe-fig-007` ×6; `qe-fig-005` ×7, +3 more.
References	N/A	no citations in this lecture.
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: 5. Lines: 226, 231, 576, 1136, 1281. Example: ax[0].set_title("points $x_1, x_2, \cdots, x_k$") at 226 and 231 puts \cdots in a plain (non-raw) string, an invalid escape sequence (pycodestyle W605) - the identical titles at 543, 547 and 551 are correctly written r"..."; grid_composition_transform(B,A) at 576 drops the space after the comma that the parallel call at 568 has; plt.quiver(*origin, - eigenvectors[0], - eigenvectors[1], ...) at 1136-1137 puts a space after the unary minus; ax.quiver(x, y, u_imag, u_real-x, v_real-y, v_imag-u_imag, ...) at 1281 omits the spaces around the binary minus that the same expression has at 1125.
[qe-fig-001] — Do not set figure size unless necessary. Count: 6. Lines: 212, 238, 530, 741, 1032, 1181. Example: figsize=.
[qe-fig-003] — No matplotlib embedded titles. Count: 7. Lines: 226, 231, 266, 278, 543, 547, 551. Example: .set_title.
[qe-fig-005] — Descriptive figure names for cross-referencing. Count: 7. Lines: 129, 730, 996, 1041, 1114, 1179, 1248. Example: code-cell figure without mystnb figure metadata.
[qe-fig-007] — Keep figure box and spines. Count: 6. Lines: 138, 221, 247, 539, 616, 746. Example: spine removal.
[qe-writing-004] — Avoid unnecessary capitalization in narrative text. Count: 5. Lines: 34, 918, 925, 939, 965. Example: mid-sentence ‘Series’.
[qe-writing-006] — Capitalize lecture titles properly. Count: 1. Lines: 858. Example: H2 Title Case: ‘The Neumann Series Lemma’ (Series, Lemma).

Medium severity#

[qe-fig-008] — Use lw=2 for line charts. Count: 3. Lines: 263, 274, 1036. Example: plot() without lw=.
[qe-writing-002] (reviewer) — Keep writing clear, concise, and valuable. Count: 3. Lines: 493, 1112, 1156. Example: line 493 “This means first apply transformation $B$ and then transformation $A$” restates 486-487 word for word in meaning; line 1112 “(This is a more advanced topic in linear algebra, please step ahead if you are comfortable with the math.)” reads as the opposite of what is meant - the reader who is comfortable is the one who should stay; line 1156 repeats 1154 with “Specifically” and adds only the superlative.
[qe-writing-003] (reviewer) — Maintain logical flow. Count: 2. Lines: 699, 854. Example: line 699 promises “We now discuss the property of A that determines this behavior” after the three spiral-in / ellipse / spiral-out experiments, but the eigenvalue sections never return to those three matrices; line 854 defers with “This is discussed further later” and gives no pointer - the connection is in fact only made inside exercise eig1_ex3 (1163-1242), so the main narrative closes without answering the question it posed.
[qe-writing-005] (reviewer) — Use bold for definitions, italic for emphasis. Count: 2. Lines: 343, 720. Example: definitions set in italics instead of bold: line 343 “is called a rotation matrix” and line 720 “we say that $\lambda$ is an eigenvalue of $A$, and $v$ is the corresponding eigenvector” - these are the two central definitions of the lecture, and the lecture bolds correctly elsewhere (spectral radius at 913).

Low severity#

[qe-fig-006] — Lowercase axis labels. Count: 1. Lines: 1299. Example: axis label Im.
[qe-writing-007] (reviewer) — Use visual elements to enhance understanding. Count: 1. Lines: 858. Example: the Neumann Series Lemma section (858-965) is the only substantive section with no figure, although its whole content is the condition $r(A) < 1$ - the natural picture, eigenvalues plotted inside the unit disk, or the truncated sum converging to $(I-A)^{-1}$, is absent even though the lecture gives every other concept a panel.

Strengths#

The transformation taxonomy is built on one reusable pair of functions, grid_transform and circle_transform (194-281), then applied unchanged to scaling, shearing, rotation and permutation - so the four sections are directly comparable panel for panel.
Non-commutativity is argued twice over: numerically at 404-484, where \underbrace / \overbrace labels tie every bmatrix to $A$, $B$, $x$ and $Bx$, and then visually with the shear-then-rotate / rotate-then-shear grids at 567 and 575.
Both labelled equations are cited: gp_sum (873) at 905 and neumann_eqn (895) at 902 - no orphan labels, and no manual “equation (1)” references anywhere.
plot_series carries its own (plot_series)= anchor at 600 and is reached by {ref}Previously <plot_series>`` from exercise eig1_ex3 at 1166, so the exercise points back at the exact cell it builds on.
Every matrix is written with bmatrix rather than array, and Greek names in code are Unicode (θ at 249, 348, 618), so qe-math-003 and qe-code-002 are clean throughout.

Recommended actions#

Move the 7 ax.set_title(...) calls (226, 231, 266, 278, 543, 547, 551) into mystnb figure captions and add mystnb: figure: name/caption metadata to the 7 plotting cells at 129, 730, 996, 1041, 1114, 1179 and 1248 - these two fixes travel together and are the largest item in the lecture (qe-fig-003 x7, qe-fig-005 x7).
Lower-case the theorem name in narrative text and headings: “Neumann Series Lemma” -> “Neumann series lemma” at 34, 858, 918, 925, 939 and 965 (qe-writing-004 x5, qe-writing-006 x1).
Bold the two definitions instead of italicising them - rotation matrix at 343 and eigenvalue / eigenvector at 720 - matching the spectral radius at 913.
Close the loop the lecture opens at 699: after the eigenvalue definitions, come back to the three matrices from 647-687 and say which spectral radius produces spiral-in, ellipse and spiral-out, rather than deferring with “discussed further later” at 854.
Drop the 6 hand-set figsize= arguments (212, 238, 530, 741, 1032, 1181), set lw=2 on the 3 default-width line plots (263, 274, 1036), and lower-case the Im axis label at 1299 (qe-fig-001 x6, qe-fig-008 x3, qe-fig-006 x1).
Fix the code slips: make the titles at 226 and 231 raw strings as 543-551 already are, restore the comma space at 576, and even out the minus-sign spacing at 1136-1137 and 1281.
Give the Neumann series section one figure - eigenvalues in the unit disk, or the partial sums of $\sum A^k$ approaching $(I-A)^{-1}$ - and rewrite the parenthetical at 1112, which currently tells comfortable readers to skip ahead.