2025-11-20

This following is a very brief summary of what happened in class on 2025-11-20.

We worked through nearly every detail for this exercise from this week's assignment. We skipped the actual computation of the Smith normal form of the matrix, but we did explain how the list of invariant factors immediately tells us the $\mathbf{Q}[x]$-module structure and allows us to write down the corresponding rational and Jordan canonical forms. We then outlined how to get the new basis required for the rational canonical form, and the modification required to find the new basis for the Jordan canonical form.^[1]

Next class we're finally moving away from modules over a PID!

Concepts

• Rational Canonical Form III - Computation
• Jordan Canonical Form I - Definition
• Jordan Canonical Form II - Computation

References

• Dummit & Foote, Abstract Algebra: Section 12.3