Page 172 - Advanced Linear Algebra
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156 Advanced Linear Algebra
)
2 The elementary divisors of 4 are
ElemDiv²4³ ~ ElemDiv²(³ r ElemDiv²)³
where the union is a multiset union; that is, we keep all duplicate
members.
The Invariant Factor Decomposition
According to Theorem 6.4, if and are cyclic submodules with relatively
:
;
prime orders, then :l ; is a cyclic submodule whose order is the product of
the orders of and . Accordingly, in the primary cyclic decomposition of 4 ,
:
;
»»µ
4 ~ ´ ºº# »» lÄlºº# Á »» µlÄl´ºº#
Á
»»lÄlºº#
Á
Á
4 4
with elementary divisors Á satisfying
~ Á Á Ä (6.3 )
Á
we can combine cyclic summands with relatively prime orders. One judicious
(
)
way to do this is to take the leftmost highest-order cyclic submodules from
each group to get
+ ~ ºº# »» l Ä l ºº# Á »»
Á
and repeat the process
Á
+ ~ ºº# »» l Ä l ºº# Á »»
»»
Á
+ ~ ºº# »» l Ä l ºº# Á
Å
Of course, some summands may be missing here since different primary
do not necessarily have the same number of summands. In any
modules 4
case, the result of this regrouping and combining is a decomposition of the form
4~ + l Ä l +
which is called an invariant factor decomposition of 4 .
For example, suppose that
4 ~ ´ºº# »» l ºº# »»µ l ´ºº# »»µ l ´ºº# »» l ºº# »» l ºº# »»µ
Á
Á
Á
Á
Á
Á
Then the resulting regrouping and combining gives
4 ~ ´ ºº# »» l ºº# »» l ºº# »» µ l ´ ºº# »» l ºº# »» µ l ´ ºº# »» µ
Á
Á
Á
Á
Á
Á
+ + +
As to the orders of the summands, referring to 6.3 , if + ( ) has order , then
, the second–highest
since the highest powers of each prime are taken for
for and so on, we conclude that
