Meetup summary

2025-08-22 - Linear algebra basics - part 2 (determinant)

  • The 3Blue1Brown video on the determinant. This gives an excellent feel for what we want out of this function at a high level.

Agenda:

Since we’re coming to linear algebra with a specific interest in its applications to probability, we will focus on matrix determinants.1 In this session, we’ll drill into the geometric interpretation of the determinant function. From here, we’ll come up with a set of properties required of a valid determinant function and then prove that two (unmotivated) formulas satisfy these constraints and are in fact unique (the Leibniz formula and Laplace expansion). Finally, we’ll make a brief mention of the implication of these formulas for diagonal or triangular matrices (we’ll come back to this idea when we talk about matrix diagonalization). If there’s time, we’ll also talk about runtime characteristics and why neither of the above is used for numerics. (In a future session, we’ll loop back to this from the algebraic side and show how to compute this more efficiently with elementary row operations.)

Notes:

We covered my “geometric principles” determinant derivation and it seemed well received. I’m hoping to write this up and post it to the site eventually for those who didn’t attend.

Essentially everything is motivated from/builds on our previous linear algebra basics, except for one piece to reach the final Leibniz formula which depends on the uniqueness of parity of any given permutation. We were able to make arguments as to the existence of at least one decomposition into transpositions of the desired parity, but skipped uniqueness as it appears to involve some annoying case work. If somebody can come up with a succinct, satisfying proof, we can cover that next time.

NOTE: We’re most likely skipping the Memorial Weekend meet-up and will cover matrix diagonalization, (semi-)definite matrices, and the spectral theorem.

Footnotes

  1. Specifically, well-formed probability distributions must be normalized (i.e., their PDFs must integrate to 1). If you apply an invertible Rn↦Rn\mathbb{R}^n \mapsto \mathbb{R}^n function to a random vector to produce a new one, that new RV must also be normalized. Similarly, if you have some arbitrary event in the original space, its image under the transformed RV must have the same probability when computed in the new coordinate space. Since these probabilities are computed as integrals, you can analyze local changes as linear transformations. Specifically, the Jacobian matrix is a linear function that shows how the function changes when you perturb any coordinate individually. The determinant shows how volumes scale under this effective linear transform, so you have to undo this scaling to preserve probabilities under the map. ↩

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