cross_product_trick#

  • Series: lecture-python.myst

  • File: lectures/cross_product_trick.md

  • Audit date: 2026-08-26

  • Corpus snapshot: e25fdf2345

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

  • Overall score: 5.5 / 10

  • Priority: HIGH

Score breakdown#

Category

Score

One-line note

Writing

3.5/10

qe-writing-006 ×2; qe-writing-005 ×2; qe-writing-003 ×2, +3 more.

Math

3/10

qe-math-002 ×52; qe-math-006 ×5; qe-math-013 (proposed) ×1.

Code

N/A

no executable code cells.

JAX

out of scope

JAX rules target lecture-jax.

Figures

N/A

no figures or plotting code.

References

N/A

no citations in this lecture.

Links

10/10

no mechanical violations detected.

Admonitions

N/A

no admonitions, exercises or solutions.

Issues#

Critical#

None found.

High severity#

  • [qe-math-002] — Use \top for transpose notation. Count: 52. Lines: 48, 63, 68, 84, 85, 86, 119, 120, 127, 128, …. Example: apostrophe transpose x_t'.

  • [qe-math-006] — Use aligned environment correctly for PDF compatibility. Count: 5. Lines: 82, 104, 118, 126, 140. Example: bare \begin{align*} display block; the corpus convention is $\( … \begin{aligned} … \)$.

  • [qe-writing-006] — Capitalize lecture titles properly. Count: 2. Lines: 33, 92. Example: H2 Title Case: ‘Undiscounted Dynamic Programming Problem’ (Dynamic, Programming, Problem).

  • [qe-writing-008] — Remove excessive whitespace between words. Count: 24. Lines: 20, 22, 27, 28, 41, 65, 75, 80, 94, 95, …. Example: 2 spaces.

Medium severity#

  • [qe-writing-002] (reviewer) — Keep writing clear, concise, and valuable. Count: 3. Lines: 77, 80, 94. Example: line 80 is a single 55-word sentence that names both 4-tuples, both value-function matrices and both policy matrices before reaching its verb (“…are related to…by”); lines 94-95 are a 58-word sentence carrying the duality claim, the existence of the transformation and the before/after covariance structure at once, and it contains the doubled word “between between”; lines 77-78 read “That the omitted matrix \(H=0\) indicates that there are no cross products…” - the sentence has no main clause and says only that H is omitted because it is zero.

  • [qe-writing-003] (reviewer) — Maintain logical flow. Count: 2. Lines: 106, 173. Example: \(F\) carries two unrelated meanings that are never distinguished: the optimal feedback matrix in the LQ half (59, 63, 87) and the measurement-noise loading matrix in the Kalman half (106, 109, 119-120), and the duality table then uses both senses in the same table - row 165 maps \(Q\) to the Kalman \(FF'\) while row 167 maps the LQ \(F\) to \(K'\); separately, the lecture ends at 173-175 with an empty {code-cell}, so a reader who follows the Algorithm section (131-151) is left with a stub where the numerical demonstration should be.

  • [qe-writing-005] (reviewer) — Use bold for definitions, italic for emphasis. Count: 2. Lines: 74, 145. Example: bold is used for emphasis rather than definition in both places it appears outside a heading: equivalent at line 74 stresses a word in an ordinary sentence, and with non-zero covariance at line 145 stresses a qualifying phrase; neither is a term being defined, so both should be italic.

Low severity#

  • [qe-math-013 (proposed)] — Reference equations via {eq}`label`. Count: 1. Lines: 133. Example: malformed {eq} reference {eq}eq:Kalman102}`.

  • [qe-writing-009 (proposed)] — Write “IID” — not “i.i.d.” or “iid”. Count: 1. Lines: 110. Example: i.i.d…

Strengths#

  • The duality table (160-168) is a genuinely useful compact device: seven rows pairing each LQ matrix with its Kalman counterpart, which is the one thing a reader will come back to this lecture for.

  • Every matrix has its dimensions declared before it is used - the 5-tuple at line 42, the hidden-Markov matrices at 109, and the state and signal dimensions at 113.

  • Notation stays plain throughout: single capitals for every matrix, no \mathcal, \mathbf or \boldsymbol anywhere, and starred names (\(A^*\), \(R^*\), \(B^*\), \(F^*\)) used consistently for the transformed problem.

  • The Overview states the transformation strategy as three ordered steps (27-29) before any algebra appears, and points at the prerequisite with a {doc} link rather than a raw URL.