Page 193 - Advanced Linear Algebra

The Structure of a Linear Operator 177

lies in the similarity class that is associated with the elementary divisors
Á
¸ ²%³¹. Hence, two matrices in the elementary divisor form of rational

canonical form lie in the same similarity class if and only if they have the same
multiset of elementary divisors. In other words, the elementary divisor form of
rational canonical form is a set of canonical forms for similarity, up to order of
blocks on the block diagonal.

Theorem 7.14 (The rational canonical form: elementary divisor version ) Let
=  be a finite-dimensional vector space and let B ² = ³ have minimal
polynomial

²%³ ~ ²%³Ä ²%³

where the ²%³ 's are distinct monic prime polynomials.

)
H
1 If is an elementary divisor basis for = , then ´ µ H is in the elementary
divisor form of rational canonical form:
Á Á Á
´ µ ~ diag4 H *´ ²%³µÁ à Á *´ ²%³µÁ à Á *´ ²%³µÁ à Á *´ Á ²%³µ5

where Á ² % ³ are the elementary divisors of . This block diagonal matrix

is called an elementary divisor version of a rational canonical form of .

)
2 Each similarity class of matrices contains a matrix in the elementary
9
I
divisor form of rational canonical form. Moreover, the set of matrices in I
that have this form is the set of matrices obtained from 4 by reordering the
block diagonal matrices. Any such matrix is called an elementary divisor
verison of a rational canonical form of .
(
3 The dimension of is the sum of the degrees of the elementary divisors of
)
=
, that is,

Á
dim²= ³ ~ deg² ³

~ ~
Example 7.1 Let be a linear operator on the vector space s 7 and suppose that

has minimal polynomial

²%³ ~ ²% c ³²% b ³

Noting that %c and ²% b ³ are elementary divisors and that the sum of the
degrees of all elementary divisors must equal , we have two possibilities:

)
1 %c Á ²% b 1³ Á % b
)

2 %c Á %c Á %c Á ²% b 1³
These correspond to the following rational canonical forms: