2025-11-14

This following is a very brief summary of what happened in class on 2025-11-14.

We first recapped the general description of the rational canonical form of a linear endomorphism $T:V\to V$ of a finite-dimensional $F$-vector space, $V$. We then proceeded to outline an algorithm to compute the invariant factors of $T$. The main work is to compute the Smith normal form, by starting with the matrix $x{I}_n-A$ and then proceeding to use basic row and column operations (over $F[x]$) to "diagonalize" this matrix into a matrix with a very specific shape.

We spent most of the class period working through an explicit example of computing the Smith normal form (and hence invariant factors) for a given $3\times 3$ matrix, $A$. We noted how the invariant factors let us immediately write down the rational canonical form matrix, $R$, for $A$.

We then briefly mentioned how one can also obtain the change-of-basis matrix that will conjugate $A$ to $R$. The process is a bit strange, but we will run through it in a few examples next class. And then we'll move on to the Jordan canonical form!

Concepts

• Rational Canonical Form I - Definition
• Rational Canonical Form II - Additional Properties
• Rational Canonical Form III - Computation

References

• Dummit & Foote, Abstract Algebra: Section 12.2