2024-11-18

This following is a very brief summary of what happened in class on 2024-11-18.

Beginning with:

• a field, $F$
• a finite-dimensional $F$-vector space, $V$
• an $F$-linear endomorphism, $T:V\to V$
  we put an $F[x]$-module structure on $V$ by letting $x$ act via $T$. We then noted that, as an $F[x]$-module, $V$ is still finitely generated. Since the ring $F[x]$ is a PID, our fundamental theorem guarantees an $F[x]$-module isomorphism of the form

for some nonzero nonunits $a_1(x),\dots ,a_m(x)\in F[x]$ with $a_1(x)\mid a_2(x)\mid \cdots \mid a_m(x)$ in $F[x]$. We noted that the free rank, $n$, had to be 0 (since $V$ is finite-dimensional as an $F$-vector space), and also noted that we can make the invariant factors $a_i(x)$ unique by insisting they are monic, i.e., each is of the form $a(x)=b_0+b_{1}x+\cdots +b_{k-1}{x}^{k-1}+x^k$.

We then proceeded to analyze the direct sum decomposition on the right. See our notes for full details, but in short we noted that:

• the last invariant factor, $a_m(x)$, is the minimal polynomial of $T$

• the product of the invariant factors is the characteristic polynomial of $T$

• for each summand $F[x]/⟨a(x)⟩$, the set $\{\bar{1},\bar{x},\dots ,{\bar{x}}^{k-1}\}$ is an $F$-vector space basis and with respect to this basis the matrix for $T$ (which acts as multiplication by $x$) is

  $$\left[\begin{array}{ccccc}0& 0& \cdots & \cdots & -b_0\\ 1& 0& \cdots & \cdots & -b_1\\ 0& 1& \cdots & \cdots & -b_2\\ ⋮& ⋮& \ddots & \cdots & ⋮\\ 0& 0& \cdots & 1& -b_{k-1}\end{array}\right]$$

This ultimately led to the rational canonical form for the linear transformation $T$.

We gave one example of how this all looks for a $6$-dimensional $\mathbf{\text{Q}}$-vector space with given invariant factors.

Next class we will see how to compute the invariant factors (and rational canonical form) of a given linear transformation.

Concepts

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

References

• Dummit & Foote: Section 12.2