# Jordan decomposition (of an endomorphism)

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 (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 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), let ${\mathcal G}$ be its 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 [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