2024-11-22

This following is a very brief summary of what happened in class on 2024-11-22.

We once again started with a finite-dimensional $F$ -vector space, $V$ , and an $F$ -linear endomorphism $T : V \to V$ . As usual, this allowed us to put an $F [x]$ -module structure on $V$ and then use our structure theorem to find an $F [x]$ -module isomorphism of the form

V ≃ F [x] / ⟨ a_{1} (x) ⟩ \oplus \dots \oplus F [x] / ⟨ a_{m} (x) ⟩,

where the $a_{i} (x)$ were the invariant factors of $T$ .

We then introduced a new assumption, namely that the field, $F$ , contains all of the roots of the invariant factors (equivalently, all of the roots of the minimal polynomial, $a_{m} (x)$ ). This assumption allows us to factor every invariant factor, $a (x)$ , into powers of linear polynomials:

a (x) = (x - λ_{1})^{α_{1}} \dots (x - λ_{k})^{α_{k}} .

By the Chinese Remainder Theorem, we then had a decomposition of the summand

F [x] / ⟨ a (x) ⟩ ≃ F [x] / ⟨ (x - λ_{1})^{α_{1}} \oplus \dots \oplus F [x] / (x - λ_{k})^{α_{k}} .

We then focused on each summand. Using the $F$ -basis ${(\overset{―}{x} - λ \cdot \overset{―}{1})^{k - 1}, \dots, \overset{―}{x} - λ \cdot \overset{―}{1}, \overset{―}{1}}$ , we noted that the action of $x$ on that summand would be represented by the matrix

[\begin{matrix} λ & 1 \\ λ & ⋱ \\ ⋱ & 1 \\ λ & 1 \\ λ \end{matrix}]

We called this matrix the Jordan block corresponding to that summand. Repeating this for every summand eventually resulted in the Jordan canonical form for the matrix $A$ .

We ran out of time before describing exactly how to compute the change-of-basis matrix, $Q$ , for which $J = Q^{- 1} A Q$ . See our notes for those details.

Minor homework update

Since we did not get to the change-of-basis computation for Jordan canonical form, there has been a minor change to Homework 8. The one problem that required computing such a matrix has been made optional.

Concepts

References

Dummit & Foote: Section 12.3