Page 192 - Advanced Linear Algebra

176 Advanced Linear Algebra

the figure could be replaced by the family of all multisets of prime power
polynomials.
The Rational Canonical Form

We are now ready to determine a set of canonical forms for similarity. Let
= that gives the primary cyclic
H
B ²= ³. The elementary divisor basis for
,
decomposition of =

= ~ ²ºº# »» l Ä l ºº# Á »»³ l Ä l ²ºº# Á »» l Ä l ºº# Á »»³
Á

is the union of the bases
8 Á Á ~ ²#Á #Á Ã Á c #³
Á
Á
Á
and so the matrix of with respect to is the block diagonal matrix
H

´ µ ~ diag ²*´ ²%³µÁ à Á *´ Á Á Á Á ²%³µ³
H ²%³µÁ à Á *´ ²%³µÁ à Á *´
with companion matrices on the block diagonal. This matrix has the following
form.
Definition A matrix is in the elementary divisor form of rational canonical
(
form if

( ~ diag4 *´ ²%³µÁ à Á *´ ²%³µ5

where the ²%³ are monic prime polynomials.

Thus, as shown in Figure 7.1, each similarity class contains at least one matrix
I
in the elementary divisor form of rational canonical form.

On the other hand, suppose that 4 is a rational canonical matrix
4 ~ diag ²*´ ²%³µÁ à Á *´ Á Á Á Á ²%³µ³
²%³µÁ à Á *´ ²%³µÁ à Á *´
of size d . Then 4 represents the matrix multiplication operator 4 under
;
;
the standard basis on - . The basis can be partitioned into blocks ; Á
corresponding to the position of each of the companion matrices on the block
diagonal of 4 . Since
´ O 4 º µ * ´ Á ² % ³ µ
; Á ~ »; Á
it follows from Theorem 7.12 that each subspace º »; is 4 -cyclic with monic
Á
order Á ² % ³ and so Theorem 7.9 implies that the multiset of elementary

divisors of 4 is ¸ Á ²%³¹ .

This shows two important things. First, any multiset of prime power
polynomials is the multiset of elementary divisors for some matrix. Second, 4