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(also Jordan canonical form, Jordan normal form)
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''also Jordan canonical form, Jordan normal form''
 
''also Jordan canonical form, Jordan normal form''
  
A square block-diagonal matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054340/j0543401.png" /> over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054340/j0543402.png" /> of the form
+
A square block-diagonal matrix $  J $
 +
over a field $  k $
 +
of the form $$
 +
J  =  \left \|
  
<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/j/j054/j054340/j0543403.png" /></td> </tr></table>
+
\begin{array}{cccc}
 +
J _ {n _{1}} ( \lambda _{1} )  &{}  &{}  &{}  \\
 +
{}  &\dots  &{}  & 0  \\
 +
0 &{}  &\dots  &{}  \\
 +
{}  &{}  &{}  &J _ {n _{s}} ( \lambda _{s} )  \\
 +
\end{array}
 +
\right \| ,
 +
$$
 +
where  $  J _{m} ( \lambda ) $
 +
is a square matrix of order  $  m $
 +
of the form $$
 +
J _{m} ( \lambda )  =   \left \|
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054340/j0543404.png" /> is a square matrix of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054340/j0543405.png" /> of the form
+
\begin{array}{cccccc}
 +
\lambda  & 1  &{}  &{}  &{}  &{}  \\
 +
{}  &\lambda  & 1  &{}  & 0 &{}  \\
 +
{}  &{}  &\dots  &{}  &{}  &{}  \\
 +
{}  &{}  &{}  &\dots  &{}  &{}  \\
 +
{}  & 0  &{}  &{}  &\lambda  & 1  \\
 +
{}  &{}  &{}  &{}  &{}  &\lambda  \\
 +
\end{array}
 +
\right \| ,
 +
$$
 +
$  \lambda \in k $ .  
 +
The matrix  $  J _{m} ( \lambda ) $
 +
is called the Jordan block of order $  m $
 +
with eigen value  $  m $ .  
 +
Every block is defined by an elementary divisor (cf. [[Elementary divisors|Elementary divisors]], see [[#References|[5]]]).
  
<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/j/j054/j054340/j0543406.png" /></td> </tr></table>
+
For an arbitrary square matrix  $  \lambda $
 +
over an algebraically closed field  $  A $
 +
there always exists a square non-singular matrix  $  k $
 +
over  $  C $
 +
such that  $  k $
 +
is a Jordan matrix (in other words,  $  C ^{-1} A C $
 +
is [[Similar matrices|similar]] over  $  A $
 +
to a Jordan matrix). This assertion is valid under weaker restrictions on  $  k $ :  
 +
For a matrix  $  k $
 +
to be similar to a Jordan matrix it is necessary and sufficient that  $  A $
 +
contains all roots of the minimum polynomial of  $  k $ .
 +
The matrix  $  A $
 +
mentioned above is called a Jordan form (or Jordan normal form) of the matrix  $  C ^{-1} A C $ .
 +
C. Jordan [[#References|[1]]] was one of the first to consider such a normal form (see also the historical survey in Chapts. VI and VII of [[#References|[2]]]).
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054340/j0543407.png" />. The matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054340/j0543408.png" /> is called the Jordan block of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054340/j05434010.png" /> with eigen value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054340/j05434011.png" />. Every block is defined by an elementary divisor (cf. [[Elementary divisors|Elementary divisors]], see [[#References|[5]]]).
+
The Jordan form of a matrix is not uniquely determined, but only up to the order of the Jordan blocks. More exactly, two Jordan matrices are similar over $  A $
 
+
if and only if they consist of the same Jordan blocks and differ only in the distribution of the blocks along the main diagonal. The number $  k $
For an arbitrary square matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054340/j05434012.png" /> over an algebraically closed field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054340/j05434013.png" /> there always exists a square non-singular matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054340/j05434014.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054340/j05434015.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054340/j05434016.png" /> is a Jordan matrix (in other words, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054340/j05434017.png" /> is similar over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054340/j05434018.png" /> to a Jordan matrix). This assertion is valid under weaker restrictions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054340/j05434019.png" />: For a matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054340/j05434020.png" /> to be similar to a Jordan matrix it is necessary and sufficient that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054340/j05434021.png" /> contains all roots of the minimum polynomial of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054340/j05434022.png" />. The matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054340/j05434023.png" /> mentioned above is called a Jordan form (or Jordan normal form) of the matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054340/j05434024.png" />. C. Jordan [[#References|[1]]] was one of the first to consider such a normal form (see also the historical survey in Chapts. VI and VII of [[#References|[2]]]).
+
of Jordan blocks of order $  C _{m} ( \lambda ) $
 
+
with eigen value $  m $
The Jordan form of a matrix is not uniquely determined, but only up to the order of the Jordan blocks. More exactly, two Jordan matrices are similar over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054340/j05434025.png" /> if and only if they consist of the same Jordan blocks and differ only in the distribution of the blocks along the main diagonal. The number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054340/j05434026.png" /> of Jordan blocks of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054340/j05434027.png" /> with eigen value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054340/j05434028.png" /> in a Jordan form of a matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054340/j05434029.png" /> is given by the formula
+
in a Jordan form of a matrix $  \lambda $
 
+
is given by the formula $  A $
<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/j/j054/j054340/j05434030.png" /></td> </tr></table>
+
$$
 
+
C _{m} ( \lambda )  =   \mathop{\rm rk}\nolimits ( A - \lambda E ) ^{m-1}
<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/j/j054/j054340/j05434031.png" /></td> </tr></table>
+
- 2 \  \mathop{\rm rk}\nolimits ( A - \lambda E ) ^{m} +
 
+
$$
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054340/j05434032.png" /> is the unit matrix of the same order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054340/j05434033.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054340/j05434034.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054340/j05434035.png" /> is the [[Rank|rank]] of the matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054340/j05434036.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054340/j05434037.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054340/j05434038.png" />, by definition.
+
where  $$
 +
+
 +
\mathop{\rm rk}\nolimits ( A - \lambda E ) ^{m+1} ,
 +
$$
 +
is the unit matrix of the same order $  E $
 +
as $  n $ ,  
 +
$  A $
 +
is the [[Rank|rank]] of the matrix $  \mathop{\rm rk}\nolimits \  B $ ,  
 +
and $  B $
 +
is $  \mathop{\rm rk}\nolimits ( A - \lambda E ) ^{0} $ ,  
 +
by definition.
  
 
There are other types of normal forms of matrices besides a Jordan normal form. They are resorted to, for example, when it is desired to avoid the non-uniqueness of the reduction to a Jordan normal form, or when the ground field does not contain all roots of the minimum polynomial of the matrix (see [[#References|[2]]]–[[#References|[5]]]).
 
There are other types of normal forms of matrices besides a Jordan normal form. They are resorted to, for example, when it is desired to avoid the non-uniqueness of the reduction to a Jordan normal form, or when the ground field does not contain all roots of the minimum polynomial of the matrix (see [[#References|[2]]]–[[#References|[5]]]).

Latest revision as of 11:11, 17 December 2019

j0543401.png ~/encyclopedia/old_files/data/J054/J.0504340 37 0 38 also Jordan canonical form, Jordan normal form

A square block-diagonal matrix $ J $ over a field $ k $ of the form $$ J = \left \| \begin{array}{cccc} J _ {n _{1}} ( \lambda _{1} ) &{} &{} &{} \\ {} &\dots &{} & 0 \\ 0 &{} &\dots &{} \\ {} &{} &{} &J _ {n _{s}} ( \lambda _{s} ) \\ \end{array} \right \| , $$ where $ J _{m} ( \lambda ) $ is a square matrix of order $ m $ of the form $$ J _{m} ( \lambda ) = \left \| \begin{array}{cccccc} \lambda & 1 &{} &{} &{} &{} \\ {} &\lambda & 1 &{} & 0 &{} \\ {} &{} &\dots &{} &{} &{} \\ {} &{} &{} &\dots &{} &{} \\ {} & 0 &{} &{} &\lambda & 1 \\ {} &{} &{} &{} &{} &\lambda \\ \end{array} \right \| , $$ $ \lambda \in k $ . The matrix $ J _{m} ( \lambda ) $ is called the Jordan block of order $ m $ with eigen value $ m $ . Every block is defined by an elementary divisor (cf. Elementary divisors, see [5]).

For an arbitrary square matrix $ \lambda $ over an algebraically closed field $ A $ there always exists a square non-singular matrix $ k $ over $ C $ such that $ k $ is a Jordan matrix (in other words, $ C ^{-1} A C $ is similar over $ A $ to a Jordan matrix). This assertion is valid under weaker restrictions on $ k $ : For a matrix $ k $ to be similar to a Jordan matrix it is necessary and sufficient that $ A $ contains all roots of the minimum polynomial of $ k $ . The matrix $ A $ mentioned above is called a Jordan form (or Jordan normal form) of the matrix $ C ^{-1} A C $ . C. Jordan [1] was one of the first to consider such a normal form (see also the historical survey in Chapts. VI and VII of [2]).

The Jordan form of a matrix is not uniquely determined, but only up to the order of the Jordan blocks. More exactly, two Jordan matrices are similar over $ A $ if and only if they consist of the same Jordan blocks and differ only in the distribution of the blocks along the main diagonal. The number $ k $ of Jordan blocks of order $ C _{m} ( \lambda ) $ with eigen value $ m $ in a Jordan form of a matrix $ \lambda $ is given by the formula $ A $ $$ C _{m} ( \lambda ) = \mathop{\rm rk}\nolimits ( A - \lambda E ) ^{m-1} - 2 \ \mathop{\rm rk}\nolimits ( A - \lambda E ) ^{m} + $$ where $$ + \mathop{\rm rk}\nolimits ( A - \lambda E ) ^{m+1} , $$ is the unit matrix of the same order $ E $ as $ n $ , $ A $ is the rank of the matrix $ \mathop{\rm rk}\nolimits \ B $ , and $ B $ is $ \mathop{\rm rk}\nolimits ( A - \lambda E ) ^{0} $ , by definition.

There are other types of normal forms of matrices besides a Jordan normal form. They are resorted to, for example, when it is desired to avoid the non-uniqueness of the reduction to a Jordan normal form, or when the ground field does not contain all roots of the minimum polynomial of the matrix (see [2][5]).

From the point of view of the theory of invariants, a Jordan matrix is a canonical representative in the orbits of the adjoint representation of the general linear group. The determination of analogous representatives for an arbitrary reductive algebraic group is still (1978) not completely solved (see [6][7]).

References

[1] C. Jordan, "Traité des substitutions et des équations algébriques" , Paris (1870) pp. 114–125 MR1188877 MR0091260 Zbl 03.0042.02
[2] N. Bourbaki, "Elements of mathematics. Algebra: Modules. Rings. Forms" , 2 , Addison-Wesley (1975) pp. Chapt.4;5;6 (Translated from French) MR0643362 Zbl 1139.12001
[3] F.R. [F.R. Gantmakher] Gantmacher, "The theory of matrices" , 1 , Chelsea, reprint (1977) (Translated from Russian) MR1657129 MR0107649 MR0107648 Zbl 0927.15002 Zbl 0927.15001 Zbl 0085.01001
[4] S. Lang, "Algebra" , Addison-Wesley (1974) MR0783636 Zbl 0712.00001
[5] A.I. Mal'tsev, "Foundations of linear algebra" , Freeman (1963) (Translated from Russian) Zbl 0396.15001
[6] A. Borel (ed.) R. Carter (ed.) C.W. Curtis (ed.) N. Iwahori (ed.) T.A. Springer (ed.) R. Steinberg (ed.) , Seminar on algebraic groups and related finite groups , Lect. notes in math. , 131 , Springer (1970) Zbl 0192.36201
[7] R. Steinberg, "Classes of elements of semisimple algebraic groups" , Internat. Congress Mathematicians (Moscow, 1966) , Mir (1968) pp. 277–283 MR0238856 Zbl 0192.36202
How to Cite This Entry:
Jordan matrix. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Jordan_matrix&oldid=33698
This article was adapted from an original article by V.L. Popov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article