Difference between revisions of "Jordan decomposition (of an endomorphism)"
Ulf Rehmann (talk | contribs) m (MR/ZBL numbers added) |
m (moved Jordan decomposition to Jordan decomposition (of an endomorphism): Disambiguation) |
(No difference)
|
Revision as of 18:30, 21 August 2012
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=21983