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''of a matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035300/e0353001.png" /> over a polynomial ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035300/e0353002.png" />''
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Powers of the monic irreducible polynomials over the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035300/e0353003.png" /> into which the invariant factors of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035300/e0353004.png" /> split. Two <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035300/e0353005.png" />-matrices over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035300/e0353006.png" /> having the same rank are equivalent (that is, can be obtained from one another by means of elementary operations) if and only if they have the same system of elementary divisors.
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The elementary divisors of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035300/e0353007.png" />-matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035300/e0353008.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035300/e0353009.png" /> are, by definition, those of its characteristic matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035300/e03530010.png" />. They can be obtained in the following manner. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035300/e03530011.png" /> be the greatest common divisor of the minors of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035300/e03530012.png" /> of the matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035300/e03530013.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035300/e03530014.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035300/e03530015.png" />. Then the invariant factors of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035300/e03530016.png" /> are
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''of a matrix $  F ( x) $
 +
over a polynomial ring  $  k [ x] $''
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035300/e03530017.png" /></td> </tr></table>
+
Powers of the monic irreducible polynomials over the field  $  k $
 +
into which the invariant factors of  $  F ( x) $
 +
split. Two  $  ( m \times n ) $-
 +
matrices over  $  k [ x] $
 +
having the same rank are equivalent (that is, can be obtained from one another by means of elementary operations) if and only if they have the same system of elementary divisors.
  
The factors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035300/e03530018.png" /> different from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035300/e03530019.png" /> are in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035300/e03530020.png" />. Each of them can be represented in the form
+
The elementary divisors of an  $  ( n \times n ) $-
 +
matrix  $  A $
 +
over  $  k $
 +
are, by definition, those of its characteristic matrix  $  \| x E _ {n} - A \| $.  
 +
They can be obtained in the following manner. Let  $  D _ {l} ( x) $
 +
be the greatest common divisor of the minors of order  $  l $
 +
of the matrix  $  \| x E _ {n} - A \| $,
 +
$  1 \leq  l \leq  n $,
 +
and let  $  D _ {0} = 1 $.  
 +
Then the invariant factors of  $  \| x E _ {n} - A \| $
 +
are
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035300/e03530021.png" /></td> </tr></table>
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$$
 +
i _ {l} ( x)  =
 +
\frac{D _ {l} ( x) }{D _ {l-} 1 ( x) }
 +
,\  i = 1 \dots n .
 +
$$
  
where the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035300/e03530022.png" /> are monic irreducible polynomials over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035300/e03530023.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035300/e03530024.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035300/e03530025.png" /> when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035300/e03530026.png" />. All the polynomials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035300/e03530027.png" /> thus obtained form the system of elementary divisors of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035300/e03530028.png" />. Two square matrices over a field are similar if and only if they have the same system of elementary divisors. The product of all elementary divisors of a matrix over a field is its characteristic polynomial, and their least common multiple is its minimum polynomial. Any collection of polynomials of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035300/e03530029.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035300/e03530030.png" /> is a monic irreducible polynomial over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035300/e03530031.png" />, is the system of elementary divisors for one and only one class of similar matrices over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035300/e03530032.png" /> of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035300/e03530033.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035300/e03530034.png" /> is the degree of the product of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035300/e03530035.png" />.
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The factors  $  i _ {l} ( x) $
 +
different from  $  1 $
 +
are in  $  k [ x] \setminus  k $.  
 +
Each of them can be represented in the form
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035300/e03530036.png" /> is a splitting field of the characteristic polynomial of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035300/e03530037.png" />, then the elementary divisors have the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035300/e03530038.png" />. Their number is then the same as the number of Jordan cells in the Jordan form of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035300/e03530039.png" />, and the elementary divisor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035300/e03530040.png" /> corresponds to a Jordan cell <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035300/e03530041.png" /> of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035300/e03530042.png" /> (see [[Jordan matrix|Jordan matrix]]). A square matrix over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035300/e03530043.png" /> is similar to a diagonal matrix if and only if each elementary divisor of it has the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035300/e03530044.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035300/e03530045.png" />.
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$$
 +
i _ {l} ( x)  = ( p _ {1} ( x) ) ^ {m _ {1} } \dots
 +
( p _ {t} ( x) ) ^ {m _ {t} } ,
 +
$$
  
 +
where the  $  p _ {i} ( x) $
 +
are monic irreducible polynomials over  $  k $,
 +
$  m _ {i} > 0 $,
 +
and  $  p _ {i} ( x) \neq p _ {j} ( x) $
 +
when  $  i \neq j $.
 +
All the polynomials  $  ( p ( x) )  ^ {m} $
 +
thus obtained form the system of elementary divisors of  $  A $.
 +
Two square matrices over a field are similar if and only if they have the same system of elementary divisors. The product of all elementary divisors of a matrix over a field is its characteristic polynomial, and their least common multiple is its minimum polynomial. Any collection of polynomials of the form  $  l _ {i} ( x) = g _ {i} ( x) ^ {m _ {i} } $,
 +
where  $  g _ {i} ( x) $
 +
is a monic irreducible polynomial over  $  k $,
 +
is the system of elementary divisors for one and only one class of similar matrices over  $  k $
 +
of order  $  n $,
 +
where  $  n $
 +
is the degree of the product of the  $  l _ {i} ( x) $.
  
 +
If  $  k $
 +
is a splitting field of the characteristic polynomial of  $  A $,
 +
then the elementary divisors have the form  $  ( x - \lambda )  ^ {m} $.
 +
Their number is then the same as the number of Jordan cells in the Jordan form of  $  A $,
 +
and the elementary divisor  $  ( x - \lambda )  ^ {m} $
 +
corresponds to a Jordan cell  $  J _ {m} ( \lambda ) $
 +
of order  $  m $(
 +
see [[Jordan matrix|Jordan matrix]]). A square matrix over a field  $  k $
 +
is similar to a diagonal matrix if and only if each elementary divisor of it has the form  $  x - \lambda $,
 +
where  $  \lambda \in k $.
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  N. Jacobson,  "Lectures in abstract algebra" , '''II. Linear algebra''' , v. Nostrand  (1953)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  N. Jacobson,  "Basic algebra" , '''I''' , Freeman  (1974)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  N. Jacobson,  "Lectures in abstract algebra" , '''II. Linear algebra''' , v. Nostrand  (1953)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  N. Jacobson,  "Basic algebra" , '''I''' , Freeman  (1974)</TD></TR></table>

Revision as of 19:37, 5 June 2020


of a matrix $ F ( x) $ over a polynomial ring $ k [ x] $

Powers of the monic irreducible polynomials over the field $ k $ into which the invariant factors of $ F ( x) $ split. Two $ ( m \times n ) $- matrices over $ k [ x] $ having the same rank are equivalent (that is, can be obtained from one another by means of elementary operations) if and only if they have the same system of elementary divisors.

The elementary divisors of an $ ( n \times n ) $- matrix $ A $ over $ k $ are, by definition, those of its characteristic matrix $ \| x E _ {n} - A \| $. They can be obtained in the following manner. Let $ D _ {l} ( x) $ be the greatest common divisor of the minors of order $ l $ of the matrix $ \| x E _ {n} - A \| $, $ 1 \leq l \leq n $, and let $ D _ {0} = 1 $. Then the invariant factors of $ \| x E _ {n} - A \| $ are

$$ i _ {l} ( x) = \frac{D _ {l} ( x) }{D _ {l-} 1 ( x) } ,\ i = 1 \dots n . $$

The factors $ i _ {l} ( x) $ different from $ 1 $ are in $ k [ x] \setminus k $. Each of them can be represented in the form

$$ i _ {l} ( x) = ( p _ {1} ( x) ) ^ {m _ {1} } \dots ( p _ {t} ( x) ) ^ {m _ {t} } , $$

where the $ p _ {i} ( x) $ are monic irreducible polynomials over $ k $, $ m _ {i} > 0 $, and $ p _ {i} ( x) \neq p _ {j} ( x) $ when $ i \neq j $. All the polynomials $ ( p ( x) ) ^ {m} $ thus obtained form the system of elementary divisors of $ A $. Two square matrices over a field are similar if and only if they have the same system of elementary divisors. The product of all elementary divisors of a matrix over a field is its characteristic polynomial, and their least common multiple is its minimum polynomial. Any collection of polynomials of the form $ l _ {i} ( x) = g _ {i} ( x) ^ {m _ {i} } $, where $ g _ {i} ( x) $ is a monic irreducible polynomial over $ k $, is the system of elementary divisors for one and only one class of similar matrices over $ k $ of order $ n $, where $ n $ is the degree of the product of the $ l _ {i} ( x) $.

If $ k $ is a splitting field of the characteristic polynomial of $ A $, then the elementary divisors have the form $ ( x - \lambda ) ^ {m} $. Their number is then the same as the number of Jordan cells in the Jordan form of $ A $, and the elementary divisor $ ( x - \lambda ) ^ {m} $ corresponds to a Jordan cell $ J _ {m} ( \lambda ) $ of order $ m $( see Jordan matrix). A square matrix over a field $ k $ is similar to a diagonal matrix if and only if each elementary divisor of it has the form $ x - \lambda $, where $ \lambda \in k $.

Comments

References

[a1] N. Jacobson, "Lectures in abstract algebra" , II. Linear algebra , v. Nostrand (1953)
[a2] N. Jacobson, "Basic algebra" , I , Freeman (1974)
How to Cite This Entry:
Elementary divisors. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Elementary_divisors&oldid=46800
This article was adapted from an original article by D.A. Suprunenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article