Math Review Hooks

Hook

Stochastic models constantly move vectors by matrices (μP, πG=0, Ax=b). Fluency here removes friction later.

Core toolkit

1. Matrix–vector product; linearity
2. Row-stochastic vs generator (row-sum 0) intuition
3. Eigen-slogan: stationary problems are eigenvector problems with special normalization
4. Series sketch: exp(tG)=∑ (tG)^k/k!

Checkpoint

1. Compute [[1,2],[3,4]](1,0)^T.

2. Invent a legal 2×2 generator G.

   Core facts

   1. Name the objects precisely — symbols before slogans.
   2. State the governing identity or update rule for this lesson.
   3. Compute one tiny example by hand before touching software.
   4. Track what assumption would break the formula.
   5. Rephrase the result in one sentence a teammate could check.

   Why this matters

   This chapter exists so you can move numbers, not just recognize vocabulary. If you cannot reproduce the worked examples with paper closed after one pass, treat the interactive lab as diagnosis, not entertainment.

   Derivation sketch

   Write the definition, apply the main theorem or algorithm step, then specialize to a two- or three-dimensional toy case. Keep units / probability normalization visible. The interactive at the end should match your hand calculation within noise.

   Worked example A

   Take the simplest nontrivial parameters from the definitions above. Compute the primary quantity twice: once symbolically, once numerically. Compare.

   Worked example B

   Change one assumption (rate↔mean, train↔test, real↔risk-neutral, classical↔quantum). Predict the direction of change before recalculating.