Rational Canonical Form II - Additional Properties

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.

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:

$S$ 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; and
the $F [x]$ -modules obtained from $V$ via $S$ and $T$ are isomorphic $F [x]$ -modules.

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

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 $A$ .