Page 198 - Advanced Linear Algebra

182 Advanced Linear Algebra

Á
Proof. Consider the companion matrix *´ ²%³µ in the rational canonical form

of over . This is a matrix over as well and Theorem 7.15 implies that
-
(
2

*´ ²%³µ — diag ²*´ Á Á Á µÁ à Á *´ Á Á µ³
Á Á
8
Hence, is an elementary divisor basis for over .
(
2
As mentioned, unlike the elementary divisors, the invariant factors are field
independent. This is equivalent to saying that the invariant factors of a matrix
( 4 ²-³ are polynomials over the smallest subfield of - that contains the

entries of (À
Theorem 7.20 Let ( C ²-³ and let , ‹ - be the smallest subfield of -
that contains the entries of .
(
1 The invariant factors of are polynomials over .
)
(
,
) are similar over if and only if they are
2 Two matrices (Á )  C ²-³ -
similar over .
,
)
Proof. Part 1 follows immediately from Theorem 7.19, since using either or
7
8 to compute invariant factors gives the same result. Part 2) follows from the
fact that two matrices are similar over a given field if and only if they have the
same multiset of invariant factors over that field.
Example 7.2 Over the real field, the matrix
c
(~ 6 7

is the companion matrix for the polynomial %b , and so

ElemDiv s InvFact s ²(³ ~ ¸% b ¹ ~ ²(³
However, as a complex matrix, the rational canonical form for is
(

(~ 6 7
c
and so

and InvFact d ²(³ ~ ¸% c Á % b ¹ ²(³ ~ ¸% b ¹
ElemDiv d
Exercises

=
1. We have seen that any B ²= ³ can be used to make into an -´%µ -
module. Does every module = over - ´ % µ come from some  B ² = ? ³
Explain.
B

2. Let ²= ³ have minimal polynomial
²%³ ~ ²%³Ä ²%³