Rational Canonical Form II - Additional Properties

The following are some additional properties of rational canonical forms.

Existence and uniqueness of rational canonical forms

Rational canonical form for linear transformations

Let $V$ be a finite-dimensional vector space over a field $F$ , and let $T : V \to V$ be a linear endomorphism. Then there is a basis $B$ for $V$ such that the matrix $M (T; B)$ for $T$ with respect to the basis $B$ is in rational canonical form.

The rational canonical form for $T$ is unique.

This is a direct corollary of the Fundamental Theorem for modules over a PID in the case of the ring $R = F [x]$ . As explained in this note, the isomorphism

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

exactly corresponds to a choice of $F$ -basis for $V$ for which the corresponding matrix for $T$ is in rational canonical form. (The uniqueness followed from insisting the invariant factors $a_{i} (x)$ be monic polynomials, and hence were no longer only unique "up to unit.")

Rational canonical form and matrix similarity

For a given $F$ -vector space $V$ , each linear endomorphism $T : V \to V$ gives rise to an $F [x]$ -module structure on $V$ , and conversely. It seems reasonable, then, to ask the question:

Question

When do two linear endomorphisms $S, T : V \to V$ give rise to the "same" (i.e., isomorphic) $F [x]$ -modules?

Fortunately, the answer is as simple as you might hope:

Rational canonical form characterizes similarity

Suppose $S, T : V \to V$ are linear endomorphisms of a finite-dimensional $F$ -vector space $V$ . Then the following are equivalent:

The $F [x]$ -modules obtained from $V$ via $S$ and $T$ are isomorphic $F [x]$ -modules.
$S$ and $T$ are similar; i.e., there is a linear automorphism $C : V \to V$ such that $S = C T C^{- 1}$ .
$S$ and $T$ have the same rational canonical form.

In the above propositions, the same statements hold if "linear transformations" is replaced with $n \times n$ matrices over $F$ , where $n = \dim_{F} (V)$ .

At some point we should add a proof of the above result. For now, let's just accept it.

Invariant factors and minimal/characteristic polynomials

In light of the above results, it is reasonable to say that if you know the invariant factors of an $n \times n$ matrix, then you must know every possible property of that matrix. In particular, you should be able to deduce things like its minimal and characteristic polynomials.

Indeed, you can. We've already noted that the minimal polynomial is the largest invariant factor. For the rest, we have the following:

Characteristic polynomials and invariant factors

Let $A$ be an $n \times n$ matrix over a field $F$ . Then:

The characteristic polynomial of $A$ is the product of the invariant factors of $A$ .
(The Cayley-Hamilton Theorem) The minimal polynomial of $A$ divides the characteristic polynomial of $A$ .
The characteristic polynomial of $A$ divides some power of the minimal polynomial of $A$ .

The first fact above should probably be the definition of the characteristic polynomial of $A$ . In any case, the second and third facts both immediately follow from the divisibility condition $a_{1} (x) ∣ a_{2} (x) ∣ \dots ∣ a_{m} (x)$ and the fact that $a_{m} (x)$ is the minimal polynomial of $A$ .

Minor note

In many textbooks, the characteristic polynomial of $A$ is defined as $p (x) = det (A - x I_{n})$ (or maybe $p (λ) = det (A - λ I_{n})$ , if the focus is on eigenvalues). In our setup, however, the characteristic polynomial is taken to be $det (x I_{n} - A)$ , which is guaranteed to be monic. In general, the only difference is a possible minus sign, since $det (A - x I_{n}) = (- 1)^{n} det (x I_{n} - A)$ .

Suggested next note

Rational Canonical Form III - Computation