Page 195 - Advanced Linear Algebra

The Structure of a Linear Operator 179

( ~ diag4 *´ ²%³µÁ à Á *´ ²%³µ5

where b ²%³ “ ²%³ for ~ ÁÃÁ c .
Theorem 7.15 can be used to rearrange and combine the companion matrices in
an elementary divisor version of a rational canonical form to produce an
9
invariant factor version of rational canonical form that is similar to . Also, this
9
process is reversible.
(
)
Theorem 7.16 The rational canonical form: invariant factor version Let
dim²= ³  B and suppose that  ²= ³ has minimal polynomial
B
²%³ ~ ²%³Ä ²%³

)
(
where the monic polynomials ²%³ are distinct prime irreducible polynomials

)
8
1 = has an invariant factor basis , that is, a basis for which
´ µ ~ diag4 8 *´ ²%³µÁ à Á *´ ²%³µ5
where the polynomials ²%³ are the invariant factors of and

² b % ³ “ ² % ³ . This block diagonal matrix is called an invariant factor
version of a rational canonical form of .

2 Each similarity class of matrices contains a matrix in the invariant
)
9
I
factor 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 invariant factor
verison of a rational canonical form of .
(
)
3 The dimension of is the sum of the degrees of the invariant factors of ,

=
that is,

dim²= ³ ~ deg² ³
~
The Determinant Form of the Characteristic Polynomial
In general, the minimal polynomial of an operator is hard to find. One of the

virtues of the characteristic polynomial is that it is comparatively easy to find.
This also provides a nice example of the theoretical value of the rational
canonical form.
Let us first take the case of a companion matrix. If (~ *´ ²%³µ is the

companion matrix of a monic polynomial
² % Â Á Ã Á ³ ~ c b % b Ä b c % c b %
then how can we recover ²%³ ~ ²%³ from *´ ²%³µ by arithmetic operations?
(