Layered hidden Markov model: Difference between revisions

Revision as of 10:01, 7 December 2012

The invariant factors of a module over a principal ideal domain (PID) occur in one form of the structure theorem for finitely generated modules over a principal ideal domain.

If $R$ is a PID and $M$ a finitely generated $R$ -module, then

M ≅ R^{r} \oplus R / (a_{1}) \oplus R / (a_{2}) \oplus \dots \oplus R / (a_{m})

for some integer $r \geq 0$ and a (possibly empty) list of nonzero elements $a_{1}, \dots, a_{m} \in R$ for which $a_{1} ∣ a_{2} ∣ \dots ∣ a_{m}$ . The nonnegative integer $r$ is called the free rank or Betti number of the module $M$ , while $a_{1}, \dots, a_{m}$ are the invariant factors of $M$ and are unique up to associatedness.

The invariant factors of a matrix over a PID occur in the Smith normal form and provide a means of computing the structure of a module from a set of generators and relations.

References

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My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 Chap.8, p.128.
Chapter III.7, p.153 of Template:Lang Algebra

Template:Abstract-algebra-stub

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+The '''invariant factors''' of a [[module (mathematics)|module]] over a [[principal ideal domain]] (PID) occur in one form of the [[structure theorem for finitely generated modules over a principal ideal domain]].
+If <math>R</math> is a [[Principal ideal domain|PID]] and <math>M</math> a [[Finitely-generated module|finitely generated]] <math>R</math>-module, then
+:<math>M\cong R^r\oplus R/(a_1)\oplus R/(a_2)\oplus\cdots\oplus R/(a_m)</math>
+for some integer <math>r\geq0</math> and a (possibly empty) list of nonzero elements <math>a_1,\ldots,a_m\in R</math> for which <math>a_1 \mid a_2 \mid \cdots \mid a_m</math>. The nonnegative integer <math>r</math> is called the ''free rank'' or ''Betti number'' of the module <math>M</math>, while <math>a_1,\ldots,a_m</math> are the ''invariant factors'' of <math>M</math> and are unique up to [[associatedness]].
+The invariant factors of a [[Matrix (mathematics)|matrix]] over a PID occur in the [[Smith normal form]] and provide a means of computing the structure of a module from a set of generators and relations.
+==See also==
+* [[Elementary divisors]]
+==References==
+* {{cite book | author=B. Hartley | authorlink=Brian Hartley | coauthors=T.O. Hawkes | title=Rings, modules and linear algebra | publisher=Chapman and Hall | year=1970 | isbn=0-412-09810-5 }}  Chap.8, p.128.
+* Chapter III.7, p.153 of {{Lang Algebra|edition=3}}
+[[Category:Module theory]]
+{{Abstract-algebra-stub}}

Layered hidden Markov model: Difference between revisions

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