Jordan decomposition (of an endomorphism)
The Jordan decomposition of a function 
of bounded variation is the representation of 
in the form
 |
where 
and 
are monotone increasing functions. A Jordan decomposition is also the representation of a signed measure or a charge 
on measurable sets 
as a difference of measures,
 |
where at least one of the measures (cf. Measure) 
and 
is finite. Established by C. Jordan.
References
| [1] | C. Jordan, "Cours d'analyse" , 1 , Gauthier-Villars (1893) MR1188188 MR1188187 MR1188186 MR0710200 |
| [2] | P.R. Halmos, "Measure theory" , v. Nostrand (1950) MR0033869 Zbl 0040.16802 |
| [3] | I.P. Natanson, "Theorie der Funktionen einer reellen Veränderlichen" , H. Deutsch , Frankfurt a.M. (1961) (Translated from Russian) MR0640867 MR0409747 MR0259033 MR0063424 Zbl 0097.26601 |
M.I. Voitsekhovskii
the Jordan decomposition of an endomorphism 
of a finite-dimensional vector space is the representation of 
as the sum of a semi-simple and a nilpotent endomorphism that commute with each other: 
. The endomorphisms 
and 
are said to be the semi-simple and the nilpotent component of the Jordan decomposition of 
. 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 
of 
is a Jordan matrix (i.e., a matrix in Jordan canonical form), and 
is an endomorphism such that the matrix 
of 
in the same basis has 
for 
and 
for all 
, then
 |
is the Jordan decomposition of 
with 
and 
.
The Jordan decomposition exists and is unique for any endomorphism 
of a vector space 
over an algebraically closed field 
. Moreover, 
and 
for some polynomials 
and 
over 
(depending on 
) with constant terms equal to zero. If 
is a 
-invariant subspace of 
, then 
is invariant under 
and 
, and
 |
is the Jordan decomposition of 
(here 
means restriction to 
). If 
is a subfield of 
and 
is rational over 
(with respect to some 
-structure on 
), then 
and 
need not be rational over 
; one may only assert that 
and 
are rational over 
, where 
is the characteristic exponent of 
(for 
, 
is 
, and for 
it is the set of all elements of 
that are purely inseparable over 
, cf. Separable extension).
If 
is an automorphism of 
, then 
is also an automorphism of 
, and
 |
where 
and 
is the identity automorphism of 
. The automorphism 
is unipotent, that is, all its eigen values are equal to one. Every representation of 
as a product of commuting semi-simple and unipotent automorphisms coincides with the representation 
already described. This representation is called the multiplicative Jordan decomposition of the automorphism 
, and 
and 
are called the semi-simple and unipotent components of 
. If 
is rational over 
, then 
and 
are rational over 
. If 
is a 
-invariant subspace of 
, then 
is invariant under 
and 
, and
 |
is the multiplicative Jordan decomposition of 
.
The concept of a Jordan decomposition can be generalized to locally finite endomorphisms of an infinite-dimensional vector space 
, that is, endomorphisms 
such that 
is generated by finite-dimensional 
-invariant subspaces. For such 
, there is one and only one decomposition of 
as a sum 
(and in the case of an automorphism, one and only one decomposition of 
as a product 
) of commuting locally finite semi-simple and nilpotent endomorphisms (semi-simple and unipotent automorphisms, respectively), that is, endomorphisms (automorphisms) such that every finite-dimensional 
-invariant subspace 
of 
is invariant under 
and 
(
and 
, respectively) and 
(
, respectively) is the Jordan decomposition of 
.
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 
be an affine algebraic group over 
(cf. Affine group), let 
be its Lie algebra, let 
be the representation of 
in the group of automorphisms of the algebra 
of regular functions on 
defined by right translations, and let 
be its derivation. For arbitrary 
in 
and 
in 
, the endomorphisms 
and 
of the vector space 
are locally finite, so that one can speak of their Jordan decompositions:
 |
and
 |
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 
and 
, respectively. More exactly, there exist unique elements 
and 
such that
 | (1) |
 | (2) |
and
 |
 |
The decomposition (1) is called the Jordan decomposition in the algebraic group 
, and (2) the Jordan decomposition in the algebraic Lie algebra 
. If 
is defined over a subfield 
of 
and the element 
(
, respectively) is rational over 
, then 
and 
(
and 
, respectively) are rational over 
. Moreover, if 
is realized as a closed subgroup of the general linear group 
of automorphisms of some finite-dimensional vector space 
(and thus 
is realized as a subalgebra of the Lie algebra of 
), then the Jordan decomposition (1) of 
coincides with the multiplicative decomposition introduced above for 
, while the decomposition (2) for 
coincides with the additive Jordan decomposition for 
(considered as endomorphisms of 
). If 
is a rational homomorphism of affine algebraic groups and 
is the corresponding homomorphism of their Lie algebras, then
 |
 |
for arbitrary 
, 
.
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 
is said to be semi-simple if 
, and unipotent if 
; an element 
is said to be semi-simple if 
and nilpotent if 
. If 
is defined over 
, then
 |
is a 
-closed subset of 
, and
 |
is a 
-closed subset of 
. In general,
 |
is not a closed set, but if 
is commutative, then 
and 
are closed subgroups and 
. The sets 
and 
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 [3].
References
| [1] | A. Borel, "Linear algebraic groups" , Benjamin (1969) MR0251042 Zbl 0206.49801 Zbl 0186.33201 |
| [2] | 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 |
| [3] | 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 |
V.L. Popov
Comments
References
| [a1] | R. Steinberg, "Conjugacy classes in algebraic groups" , Lect. notes in math. , 366 , Springer (1974) MR0352279 Zbl 0281.20037 |
Jordan decomposition (of an endomorphism). Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Jordan_decomposition_(of_an_endomorphism)&oldid=27713