View source for Jordan decomposition (of an endomorphism)

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The Jordan decomposition of an endomorphism  $  g $
of a finite-dimensional vector space is the representation of  $  g $
as the sum of a semi-simple and a nilpotent endomorphism that commute with each other:  $  g = g _{s} + g _{n} $. 
The endomorphisms  $  g _{s} $
and  $  g _{n} $
are said to be the semi-simple and the nilpotent component of the Jordan decomposition of  $  g $. 
This decomposition is called the additive Jordan decomposition. (A semi-simple endomorphism is one having a basis of eigen vectors for some extension of the ground field, a nilpotent endomorphism is one some power of which is the zero endomorphism.) If in some basis of the space the matrix  $  \| a _{ij} \| $
of  $  g $
is a [[Jordan matrix|Jordan matrix]] (i.e., a matrix in Jordan canonical form), and  $  t $
is an endomorphism such that the matrix  $  \| b _{ij} \| $
of  $  t $
in the same basis has  $  b _{ij} = 0 $
for  $  i \neq j $
and  $  b _{ii} = a _{ii} $
for all  $  i $, 
then

$$ 
g \  = \  t + ( g - t )
$$

is the Jordan decomposition of  $  g $
with  $  g _{s} = t $
and  $  g _{n} = g -t $.

The Jordan decomposition exists and is unique for any endomorphism  $  g $
of a vector space  $  V $
over an algebraically closed field  $  K $. 
Moreover,  $  g _{s} = P (g) $
and  $  g _{n} = Q (g) $
for some polynomials  $  P $
and  $  Q $
over  $  K $(
depending on  $  g $) 
with constant terms equal to zero. If  $  W $
is a  $  g $-
invariant subspace of  $  V $, 
then  $  W $
is invariant under  $  g _{s} $
and  $  g _{n} $, 
and

$$ 
g \mid  _{W} \  = \ 
\left . g _{s} \right | _{W} + \left . g _{n} \right | _{W}  $$

is the Jordan decomposition of  $  g \mid  _{W} $(
here  $  \mid  _{W} $
means restriction to  $  W $). 
If  $  k $
is a subfield of  $  K $
and  $  g $
is rational over  $  k $(
with respect to some  $  k $-
structure on  $  V $), 
then  $  g _{s} $
and  $  g _{n} $
need not be rational over  $  k $; 
one may only assert that  $  g _{s} $
and  $  g _{n} $
are rational over  $  k ^ {p ^ {- \infty}} $, 
where  $  p $
is the characteristic exponent of  $  k $(
for  $  p = 1 $, 
$  k ^ {p ^ {- \infty}} $
is  $  k $, 
and for  $  p > 1 $
it is the set of all elements of  $  K $
that are [[Purely inseparable extension|purely inseparable]] over  $  k $.

If  $  g $
is an automorphism of  $  V $, 
then  $  g _{s} $
is also an automorphism of  $  V $, 
and

$$ 
g = g _{s} g _{u} \  = \  g _{u} g _{s} ,
$$

where  $  g _{u} = 1 _{V} + g _ s^{-1} g _{n} $
and  $  1 _{V} $
is the identity automorphism of  $  V $. 
The automorphism  $  g _{u} $
is unipotent, that is, all its eigen values are equal to one. Every representation of  $  g $
as a product of commuting semi-simple and unipotent automorphisms coincides with the representation  $  g = g _{s} g _{u} = g _{u} g _{s} $
already described. This representation is called the multiplicative Jordan decomposition of the automorphism  $  g $, 
and  $  g _{s} $
and  $  g _{u} $
are called the semi-simple and unipotent components of  $  g $. 
If  $  g $
is rational over  $  k $, 
then  $  g _{s} $
and  $  g _{u} $
are rational over  $  k ^ {p ^ {- \infty}} $. 
If  $  W $
is a  $  g $-
invariant subspace of  $  V $, 
then  $  W $
is invariant under  $  g _{s} $
and  $  g _{u} $, 
and

$$ 
g \mid  _{W} \  = \ 
\left . g _{s} \right | _{W} \left . g _{u} \right | _{W}  $$

is the multiplicative Jordan decomposition of  $  g \mid  _{W} $.

The concept of a Jordan decomposition can be generalized to locally finite endomorphisms of an infinite-dimensional vector space  $  V $, 
that is, endomorphisms  $  g $
such that  $  V $
is generated by finite-dimensional  $  g $-
invariant subspaces. For such  $  g $, 
there is one and only one decomposition of  $  g $
as a sum  $  g = g _{s} + g _{n} $(
and in the case of an automorphism, one and only one decomposition of  $  g $
as a product  $  g _{s} g _{u} $) 
of commuting locally finite semi-simple and nilpotent endomorphisms (semi-simple and unipotent automorphisms, respectively), that is, endomorphisms (automorphisms) such that every finite-dimensional  $  g $-
invariant subspace  $  W $
of  $  V $
is invariant under  $  g _{s} $
and  $  g _{n} $(
$  g _{s} $
and  $  g _{u} $, 
respectively) and  $  g | _{W} = g _{s} \mid  _{W} + g _{n} | _{W} $(
$  g \mid  _{W} = g _{s} | _{W} g _{u} | _{W} $, 
respectively) is the Jordan decomposition of  $  g \mid  _{W} $.

This extension of the concept of a Jordan decomposition allows one to introduce the concept of a Jordan decomposition in algebraic groups and algebras. Let  $  G $
be an affine algebraic group over  $  K $(
cf. [[Affine group|Affine group]]), let  $  {\mathcal G} $
be its [[Lie algebra|Lie algebra]], let  $  \rho $
be the representation of  $  G $
in the group of automorphisms of the algebra  $  K [ G ] $
of regular functions on  $  G $
defined by right translations, and let  $  d \rho $
be its derivation. For arbitrary  $  g $
in  $  G $
and  $  X $
in  $  {\mathcal G} $, 
the endomorphisms  $  \rho (g) $
and  $  d \rho (X) $
of the vector space  $  K [ G ] $
are locally finite, so that one can speak of their Jordan decompositions:

$$ 
\rho (g) \  = \  \rho (g) _{s} \rho (g) _{u}  $$

and

$$ 
d \rho (X) \  = \  d \rho (X) _{s} + d \rho (X) _{n} .
$$

One of the important results in the theory of algebraic groups is that the Jordan decomposition just indicated is realized by the use of elements of  $  G $
and  $  {\mathcal G} $, 
respectively. More exactly, there exist unique elements  $  g _{s} ,\  g _{u} \in G $
and  $  X _{s} ,\  X _{n} \in {\mathcal G} $
such that

$$ \tag{1}
g \  = \  g _{s} g _{u} \  = \  g _{u} g _{s} ,
$$

$$ \tag{2}
X \  = \  X _{s} + X _{n} ,\ \  [ X _{s} ,\  X _{n} ] \  = \  0 ,
$$

and

$$ 
\rho ( g _{s} ) \  = \  \rho (g) _{s} ,\ \ 
\rho ( g _{u} ) \  = \  \rho (g) _{u} ,
$$

$$ 
d \rho ( X _{s} ) \  = \  ( d \rho (X) ) _{s} ,\ \  d \rho ( X _{n} ) \  = \  ( d \rho (X) ) _{n} .
$$

The decomposition (1) is called the Jordan decomposition in the algebraic group  $  G $, 
and (2) the Jordan decomposition in the algebraic Lie algebra  $  {\mathcal G} $. 
If  $  G $
is defined over a subfield  $  k $
of  $  K $
and the element  $  g \in G $(
$  X \in {\mathcal G} $, 
respectively) is rational over  $  k $, 
then  $  g _{s} $
and  $  g _{u} $(
$  X _{s} $
and  $  X _{n} $, 
respectively) are rational over  $  k ^ {p ^ {- \infty}} $. 
Moreover, if  $  G $
is realized as a closed subgroup of the general linear group  $   \mathop{\rm GL}\nolimits (V) $
of automorphisms of some finite-dimensional vector space  $  V $(
and thus  $  {\mathcal G} $
is realized as a subalgebra of the Lie algebra of  $   \mathop{\rm GL}\nolimits (V) $), 
then the Jordan decomposition (1) of  $  g \in G $
coincides with the multiplicative decomposition introduced above for  $  g $, 
while the decomposition (2) for  $  X \in {\mathcal G} $
coincides with the additive Jordan decomposition for  $  X $(
considered as endomorphisms of  $  V $). 
If  $  \phi : \  G _{1} \rightarrow G _{2} $
is a rational homomorphism of affine algebraic groups and  $  d \phi : \  {\mathcal G} _{1} \rightarrow {\mathcal G} _{2} $
is the corresponding homomorphism of their Lie algebras, then

$$ 
\phi ( g _{s} ) \  = \  \phi (g) _{s} ,\ \ 
\phi ( g _{u} ) \  = \  \phi (g) _{u} ,
$$

$$ 
d \phi ( X _{s} ) \  = \  ( d \phi (X) ) _{s} ,\ \  d \phi ( X _{n} ) \  = \  ( d \phi (X) ) _{n}  $$

for arbitrary  $  g \in G _{1} $, 
$  X \in {\mathcal G} _{1} $.

The concept of a Jordan decomposition in algebraic groups and algebraic Lie algebras allows one to introduce the definitions of a semi-simple and a unipotent (nilpotent, respectively) element in an arbitrary affine algebraic group (algebraic Lie algebra, respectively). An element  $  g \in G $
is said to be semi-simple if  $  g = g _{s} $, 
and unipotent if  $  g = g _{u} $; 
an element  $  X \in {\mathcal G} $
is said to be semi-simple if  $  X = X _{s} $
and nilpotent if  $  X = X _{n} $. 
If  $  G $
is defined over  $  k $, 
then

$$ 
G _{u} \  = \  \{ {g \in G} : {g = g _ u} \}
$$

is a  $  k $-
closed subset of  $  G $, 
and

$$ 
{\mathcal G} _{n} \  = \  \{ {X \in {\mathcal G}} : {X = X _ n} \}
$$

is a  $  k $-
closed subset of  $  {\mathcal G} $. 
In general,

$$ 
G _{s} \  = \  \{ {g \in G} : {g = g _ s} \}
$$

is not a closed set, but if  $  G $
is commutative, then  $  G _{s} $
and  $  G _{u} $
are closed subgroups and  $  G = G _{s} \times G _{u} $. 
The sets  $  G _{s} $
and  $  G _{u} $
in an arbitrary affine algebraic group are invariant under inner automorphisms, and the study of decompositions of these sets into classes of conjugate elements is a subject of special investigations [[#References|[3]]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> A. Borel, "Linear algebraic groups" , Benjamin (1969) {{MR|0251042}} {{ZBL|0206.49801}} {{ZBL|0186.33201}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> E.R. Kolchin, "Algebraic matrix groups and the Picard–Vessiot theory of homogeneous linear ordinary differential equations" ''Ann. of Math.'' , '''49''' (1948) pp. 1–42 {{MR|}} {{ZBL|}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> 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) {{MR|}} {{ZBL|0192.36201}} </TD></TR></table>

''V.L. Popov''

====Comments====

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> R. Steinberg, "Conjugacy classes in algebraic groups" , ''Lect. notes in math.'' , '''366''' , Springer (1974) {{MR|0352279}} {{ZBL|0281.20037}} </TD></TR></table>