Page 184 - Advanced Linear Algebra
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168 Advanced Linear Algebra
)
2 The elementary divisors and invariant factors of a matrix are the
(
elementary divisors and invariant factors, respectively, of the multiplication
:
operator (
ElemDiv²(³ ~ ElemDiv² ( ³ and InvFact²(³ ~ InvFact² ( ³
We emphasize that the elementary divisors and invariant factors of an operator
or matrix are monic by definition. Thus, we no longer need to worry about
uniqueness up to associate.
=
Theorem 7.6 (The primary cyclic decomposition theorem for =³ Let be
finite-dimensional and let ²= ³ have minimal polynomial
B
²%³ ~ ²%³Ä ²%³
where the polynomials ²%³ are distinct monic primes.
1 )(Primary decomposition ) The -´%µ -module = is the direct sum
=~ = l Ä l =
where
²%³
= ~ = ~ ¸# = ² ³²#³ ~ ¹
²%³
=
=
is a primary submodule of of order ² % ³ . In vector space terms, is a
= -invariant subspace of and the minimal polynomial of O is
=
min²O ³ ~ ²%³
=
2 )(Cyclic decomposition ) Each primary summand = can be decomposed
into a direct sum
= ~ ºº# »» l Ä l ºº# Á »»
Á
Á
of -cyclic submodules ºº# »» of order ²%³ with
Á
~ Á Á Ä
Á
In vector space terms, ºº# »» is a -cyclic subspace of = and the minimal
Á
is
polynomial of O ºº# »»
Á
Á
³ ~ ²%³
min²O ºº# »»
Á
3 )(The complete decomposition ) This yields the decomposition of = into a
direct sum of -cyclic subspaces
»»³
= ~ ²ºº# »» lÄlºº# Á »»³lÄl²ºº# Á »»lÄlºº# Á
Á
4 )(Elementary divisors and dimensions ) The multiset of elementary divisors
Á
¸ ²%³¹ is uniquely determined by . If deg ² ²%³³ ~ Á , then the - Á
