cross_product_trick

• Series: lecture-dp

• File: lectures/cross_product_trick.md

• Audit date: 2026-08-26

• Corpus snapshot: c30490a2f4

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

• Overall score: 5.7 / 10

• Priority: HIGH

Score breakdown

Category	Score	One-line note
Writing	4/10	qe-writing-006 ×2; qe-writing-005 ×2; qe-writing-002 ×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: 2. Lines: 80, 94. Example: the two sentences that carry the lecture’s main claims are also its longest. Line 80 is a single 55-word sentence naming two 4-tuples, two matrix pairs and the relation between them before it reaches its verb (‘are related to … by’); lines 94-95 run ~60 words and contain the duplicated word ‘between between’ plus the typo ‘measurments’. Both want splitting into one clause per sentence.

• [qe-writing-005] (reviewer) — Use bold for definitions, italic for emphasis. Count: 2. Lines: 74, 145. Example: bold used for emphasis where italic is wanted: line 74 ‘an equivalent problem without cross-products’ and line 145 ‘the original problem with non-zero covariance between shocks’. Neither is a definition - both are stressing a contrast with the preceding paragraph. The bold on duality at 94 is correct, since that sentence is where the term enters the lecture.

Low severity

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

• [qe-writing-003] (reviewer) — Maintain logical flow. Count: 1. Lines: 135. Example: the algorithm’s first step contradicts the formula it points at. Lines 135-136 say to ‘compute \(\Sigma, K^*\) using the ordinary Kalman filtering formula with \(BF' = 0\)’, but the displayed system at 140-143 is not the ordinary formula with \(BF'=0\) - it is the ordinary formula with \(A\) replaced by \(A^*\) and \(BB'\) by \(B^*{B^*}'\), the transformed matrices defined at 126-129. A reader who follows 135 literally substitutes into 119-120 and gets \(A\), not \(A^*\), so the step as written does not produce the state it claims.

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

Strengths

• The duality table at 160-168 pairs each LQ matrix with its Kalman counterpart (\(A \leftrightarrow A'\), \(H \leftrightarrow FB'\), \(P \leftrightarrow \Sigma\)), so the two halves of the lecture are explicitly tied together instead of being left as two parallel derivations the reader has to align.

• The LQ transformation at 82-88 is given as a complete five-line system - \(A^*\), \(R^*\), the Riccati equation in the starred matrices, \(F^*\), and the recovery \(F = F^* + Q^{-1}H\) - so a reader has every piece needed to implement it without going elsewhere.

• Conformability is stated for every matrix as it is introduced (line 42 for the LQ 5-tuple, line 109 for the hidden Markov model), which makes the transposes in the displayed formulas checkable at a glance.

• The forward pointer at line 28 uses the {doc} form with an explicit title ({doc}Linear Control: Foundations ``) rather than a bare URL, and it names exactly the lecture whose no-cross-product formulas the transformation is designed to reach.

Recommended actions

1. Replace the 52 apostrophe transposes with ^\top (qe-math-002) - at 52 occurrences in a 175-line file this is the dominant fix, and it touches nearly every display block plus the duality table at 162-167.

2. Convert the 5 bare \begin{align*} blocks at 82, 104, 118, 126 and 140 into $$ ... \begin{aligned} ... \end{aligned} ... $$ (qe-math-006, build risk). While doing 118-121, note that the label is currently attached as \end{align*} (eq:Kalman102), which will not survive the conversion - move it to the $$ ... $$ (eq:Kalman102) form.

3. Fix the malformed cross-reference at line 133: {eq}`eq:Kalman102} closes with a brace instead of a backtick, so the reference does not resolve (qe-math-013 (proposed)).

4. Reconcile the algorithm step at 135-136 with the formulas at 140-143 - say that \(A^*\) and \(B^*{B^*}'\) from 126-129 are substituted into the ordinary formula, rather than that \(BF'\) is set to zero.

5. Sentence-case the two H2 headings at 33 and 92 (‘Undiscounted dynamic programming problem’, ‘Kalman filter’) and write IID at line 110 in place of ‘i.i.d.’ (qe-writing-006 x2, qe-writing-009 (proposed) proposed x1).

6. Break the 55-60 word sentences at 80 and 94 into one idea each, and fix the four typos along the way: ‘between between’ (95), ‘measurments’ (95), ‘distibuted’ (110), ‘tranformed’ (124) and ‘reconstrution’ (151).

7. Clear the 24 double spaces (qe-writing-008), switch the emphasis bolds at 74 and 145 to italic, and delete the empty {code-cell} at 173-175, which renders as a blank input cell at the foot of the published page.