Page 197 - Advanced Linear Algebra
P. 197
The Structure of a Linear Operator 181
Lemma 7.17 For any ²%³ -´%µ ,
det²%0 c *´ ²%³µ³ ~ ²%³
Now suppose that 9 is a matrix in the elementary divisor form of rational
canonical form. Since the determinant of a block diagonal matrix is the product
of the determinants of the blocks on the diagonal, it follows that
det²%0 c 9³ ~ ²%³ ~ ²%³
Á
9
Á
Moreover, if ( 9 , say (~ 797 c , then
det²%0 c (³ ~ det²%0 c 797 c ³
~ det ´ 7 ² % 0 c 9 ³ 7 c µ
~ det² 7 ³ det² % 0 c 9 ³ det² 7 c ³
~ det² % 0 c 9 ³
and so
det²%0 c (³ ~ det²%0 c 9³ ~ ²%³ ~ ²%³
(
9
Hence, the fact that all matrices have a rational canonical form allows us to
deduce the following theorem.
B
(
Theorem 7.18 Let ²= ³ . If is any matrix that represents , then
²%³ ~ ²%³ ~ det ²%0 c (³
(
Changing the Base Field
A change in the base field will generally change the primeness of polynomials
and therefore has an effect on the multiset of elementary divisors. It is perhaps a
surprising fact that a change of base field has no effect on the invariant factors—
hence the adjective invariant .
Theorem 7.19 Let and be fields with - 2 . Suppose that the elementary
2
-
divisors of a matrix ( C ²-³ are
Á Á
7 ~ ¸ Á ÃÁ ÁÃÁ Á ÁÃÁ Á ¹
Suppose also that the polynomials can be further factored over , say
2
~ Á Ä Á
Á
Á
2
where Á is prime over . Then the prime powers
Á Á Á Á Á Á
Á
Á
8 ~ ¸ Á Á ÃÁ Á ÁÃÁÃÁ Á ÁÃÁ Á ¹
are the elementary divisors of over .
2
(
