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Jordan decomposition

In linear algebra, the Jordan–Chevalley decomposition is a fundamental theorem asserting that every endomorphism of a finite-dimensional vector space over an algebraically closed field admits a unique decomposition as the sum of a semisimple endomorphism and a nilpotent endomorphism that commute with each other.^[1] This decomposition, often denoted g = s + n where s is semisimple and n is nilpotent with sn = ns, provides a canonical way to separate the "diagonalizable" and "nilpotent" behaviors of linear operators.^[2] A linear endomorphism is semisimple if its minimal polynomial factors into distinct linear factors over the base field, equivalently, if it is diagonalizable in some basis.^[1] An endomorphism is nilpotent if some positive power of it is the zero map.^[1] The uniqueness of the decomposition ensures that the semisimple and nilpotent parts are well-defined invariants of the operator, independent of the choice of basis.^[3] In the context of matrices, if a matrix A is similar to its Jordan canonical form J, the semisimple part corresponds to the diagonal matrix of eigenvalues, while the nilpotent part captures the off-diagonal structure of the Jordan blocks.^[1] The theorem was originally developed by Camille Jordan in the late 19th century for matrices over the complex numbers and later generalized by Claude Chevalley to broader settings, including Lie algebras and algebraic groups, with early motivations from Hermann Weyl's work on infinitesimal transformations in the 1930s.^[3] Beyond linear algebra, the Jordan–Chevalley decomposition extends naturally to elements of Lie algebras over fields of characteristic zero, where every element decomposes uniquely into commuting semisimple and nilpotent parts within the algebra generated by the element.^[3] This structure is crucial in representation theory, the classification of semisimple Lie algebras, and the study of reductive groups, as it preserves under homomorphisms and adjoint actions.^[2] The term "Jordan decomposition" also refers to the Hahn-Jordan decomposition of signed measures in measure theory.

In linear algebra

Definition and basic properties

In linear algebra, the Jordan decomposition provides a canonical way to express a linear endomorphism T on a finite-dimensional vector space V over an algebraically closed field k as the sum of a semisimple part and a nilpotent part. Specifically, there exist unique endomorphisms S, N \in \text{End}(V) such that T = S + N, where S is semisimple, N is nilpotent, and S and N commute (i.e., SN = NS).^[3]^[1] An endomorphism S is semisimple if it is diagonalizable, meaning V admits a basis consisting entirely of eigenvectors of S.^[4] Equivalently, the minimal polynomial of S factors into distinct linear terms over k.^[4] An endomorphism N is nilpotent if there exists a positive integer m (depending on N) such that N^m = 0.^[3] The commutativity SN = NS ensures that the decomposition respects the algebraic structure of T, as both S and N are polynomials in T.^[3] This decomposition is intimately connected to the primary decomposition of V with respect to the minimal polynomial of T. If the distinct eigenvalues of T are \lambda_1, \dots, \lambda_r, then V = \bigoplus_{i=1}^r \ker((T - \lambda_i I)^{\dim V}), where each summand is the generalized eigenspace for \lambda_i. The semisimple part S is defined by its action on these spaces: S restricts to multiplication by \lambda_i on the i-th generalized eigenspace, while N = T - S restricts to a nilpotent operator on each such space.^[5] For example, consider the nilpotent endomorphism on k^2 represented by the matrix \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix} with respect to the standard basis. Here, the Jordan decomposition is S = 0 (the zero endomorphism, which is semisimple) and N equal to the original endomorphism itself (which satisfies N^2 = 0).^[3] In contrast, if T is diagonalizable, such as the diagonal matrix \begin{pmatrix} 1 & 0 \\ 0 & 2 \end{pmatrix}, then N = 0 and S = T.^[4] The Jordan normal form represents this decomposition in a basis where T appears as a block-diagonal matrix with Jordan blocks, each corresponding to the action of S and N on generalized eigenspaces.^[5]

Existence and uniqueness

The Jordan decomposition exists for any linear operator T on a finite-dimensional vector space V over an algebraically closed field k, where the characteristic polynomial of T splits completely into linear factors, ensuring all eigenvalues lie in k.^[3] To prove existence, apply the primary decomposition theorem, which decomposes V as a direct sum of the generalized eigenspaces E_\lambda = \ker((T - \lambda I)^{m_\lambda}) over the distinct eigenvalues \lambda of T, where m_\lambda is the algebraic multiplicity of \lambda.^[3] On each E_\lambda, the restriction T|_{E_\lambda} = \lambda I + N_\lambda, where N_\lambda = (T - \lambda I)|_{E_\lambda} is nilpotent, as its minimal polynomial is a power of (x)^{d} for some d.^[3] The projections P_\lambda onto E_\lambda along the direct sum of the other E_\mu (\mu \neq \lambda) are polynomials in T, given by the Chinese remainder theorem applied to the minimal polynomial factors (x - \lambda)^{m_\lambda} and \prod_{\mu \neq \lambda} (x - \mu)^{m_\mu}.^[3] Define the semisimple operator S = \sum_\lambda \lambda P_\lambda, which is diagonalizable with eigenvalues \lambda on E_\lambda and zero elsewhere, hence semisimple.^[3] The nilpotent part is then N = T - S, which restricts to N_\lambda on each E_\lambda and thus satisfies N^{m} = 0 for sufficiently large m, while S and N commute since the P_\lambda commute with T. For uniqueness, suppose T = S' + N' is another such decomposition, with S' semisimple, N' nilpotent, and S' N' = N' S'. Then S - S' = N' - N, which is both semisimple (as a difference of semisimple operators) and nilpotent (as a difference of nilpotents, but more precisely, since it commutes with T and shares the generalized eigenspace decomposition).^[3] A nonzero semisimple operator cannot be nilpotent, as its minimal polynomial has distinct roots, contradicting nilpotency unless S - S' = 0.^[3] Thus, S' is the unique semisimple operator with the same eigenvalues as T on the same generalized eigenspaces.^[3] This decomposition was first established by Camille Jordan in his 1870 treatise Traité des substitutions et des équations algébriques, building on his earlier investigations into rational canonical forms and the structure of linear substitutions.^[6]

Connection to Jordan normal form

The Jordan normal form provides a concrete matrix representation of the Jordan decomposition for a linear operator on a finite-dimensional complex vector space. Specifically, any such operator T is similar to a block diagonal matrix consisting of Jordan blocks, where each block J_k(\lambda) is of the form \lambda I_k + N_k, with I_k the k \times k identity matrix and N_k the k \times k nilpotent Jordan shift matrix (with 1s on the superdiagonal and zeros elsewhere). This structure directly reflects the semisimple-nilpotent decomposition T = S + N, where S is the semisimple part (diagonalizable, corresponding to the diagonal entries \lambda I) and N is the nilpotent part (corresponding to the superdiagonal 1s).^[7]^[8] The explicit form of a Jordan block of size k with eigenvalue \lambda is

J_k(\lambda) = \begin{pmatrix} \lambda & 1 & 0 & \cdots & 0 \\ 0 & \lambda & 1 & \cdots & 0 \\ \vdots & \vdots & \ddots & \ddots & \vdots \\ 0 & 0 & \cdots & \lambda & 1 \\ 0 & 0 & \cdots & 0 & \lambda \end{pmatrix},

and the full Jordan normal form is the direct sum of such blocks for each eigenvalue, with block sizes determined by the structure of the generalized eigenspaces. The construction proceeds by selecting a basis of generalized eigenvectors for each eigenvalue \lambda, where the dimensions of the kernels \ker((T - \lambda I)^m) for increasing m dictate the number and sizes of the blocks: the number of blocks equals the dimension of the eigenspace \ker(T - \lambda I), and larger blocks arise from higher ranks in the nilpotent part.^[7]^[3] For example, consider the $3 \times 3 matrix

A = \begin{pmatrix} -1 & -1 & 0 \\ 0 & -1 & -2 \\ 0 & 0 & -1 \end{pmatrix},

which has a repeated eigenvalue \lambda = -1 with algebraic multiplicity 3 but geometric multiplicity 1 (eigenspace dimension 1). Its Jordan normal form is

J = \begin{pmatrix} -1 & 1 & 0 \\ 0 & -1 & 1 \\ 0 & 0 & -1 \end{pmatrix},

comprising one $3 \times 3 block, illustrating how the semisimple part captures the eigenvalues on the diagonal while the nilpotent part introduces the off-diagonal structure in the larger block. This form is achieved via a change of basis using generalized eigenvectors, confirming the unique semisimple-nilpotent split.^[9]^[10]

In Lie algebras

Definition for Lie algebra elements

In the context of Lie algebras, the Jordan-Chevalley decomposition, also known as the Jordan decomposition for Lie algebra elements, provides a canonical way to express any element as a sum of a semisimple part and a nilpotent part that commute under the Lie bracket. Specifically, for a finite-dimensional semisimple Lie algebra \mathfrak{g} over an algebraically closed field of characteristic zero, every element x \in \mathfrak{g} admits a unique decomposition x = s + n, where s is semisimple (meaning the adjoint operator \mathrm{ad}_s: \mathfrak{g} \to \mathfrak{g} is diagonalizable), n is nilpotent (meaning \mathrm{ad}_n is nilpotent), and [s, n] = 0.^[11]^[12] This decomposition extends the classical Jordan decomposition from linear algebra, where it applies to endomorphisms via the adjoint representation \mathrm{ad}: \mathfrak{[g](/page/G)} \to \mathrm{End}(\mathfrak{[g](/page/G)}), but incorporates the additional compatibility condition with the Lie bracket to ensure the parts behave appropriately under the algebra's structure. It holds particularly well for semisimple Lie algebras, where \mathfrak{[g](/page/G)} is completely reducible as a module under the adjoint action \mathrm{Ad} of the corresponding Lie group, and the center of \mathfrak{[g](/page/G)} vanishes.^[11]^[13] A key property is that, in a semisimple Lie algebra, the semisimple part s lies in a Cartan subalgebra (a maximal toral subalgebra consisting of semisimple elements). The decomposition is unique, independent of the choice of basis or representation, reflecting the intrinsic nature of the adjoint operators' spectral properties.^[11]^[12] For a concrete illustration in the semisimple Lie algebra \mathfrak{sl}(2, \mathbb{C}), consider the element h = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}, which is semisimple since \mathrm{ad}_h has eigenvalues $0, 2, -2 and is thus diagonalizable. In contrast, the element f = \begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix} is nilpotent, as \mathrm{ad}_f^3 = 0 but \mathrm{ad}_f \neq 0, and the bracket satisfies [h, f] = -2f, highlighting their roles in the standard \mathfrak{sl}(2, \mathbb{C})-triple structure.^[13]^[11]

Uniqueness and commutativity

In semisimple Lie algebras over an algebraically closed field of characteristic zero, the Jordan-Chevalley decomposition of any element x \in \mathfrak{g} is unique. Specifically, if x = s + n = s' + n' where s, s' are semisimple, n, n' are nilpotent, and [s, n] = [s', n'] = 0, then s = s' and n = n'.^[11] This uniqueness follows from the fact that the center of a semisimple Lie algebra is trivial, ensuring that any difference s - s' = n' - n must be both semisimple and nilpotent, hence zero; the proof relies on the compatibility of the abstract decomposition with the adjoint representation, where the classical Jordan decomposition in \mathfrak{gl}(\mathfrak{g}) induces the unique abstract one.^[3] Semisimple elements in such algebras are those whose adjoint action is diagonalizable and can be conjugated into a Cartan subalgebra, reinforcing the uniqueness via the structure theory of root decompositions.^[14] The commutativity condition [s, n] = 0 in the decomposition plays a central role, implying that \mathrm{ad}_s and \mathrm{ad}_n commute as operators on \mathfrak{g}. Consequently, \mathrm{ad}_s preserves the generalized eigenspaces of \mathrm{ad}_n, as commuting operators share invariant subspaces, including the kernels of powers of (\mathrm{ad}_n - \lambda I).^[3] This preservation property is derived from the representation theory of \mathfrak{sl}(2, \mathbb{C}), where the decomposition aligns with irreducible representations generated by sl(2)-triples associated to root vectors, ensuring the nilpotent part acts appropriately on weight spaces while the semisimple part acts diagonally.^[14] The Jordan-Chevalley decomposition extends uniquely to reductive Lie algebras, which decompose as a direct sum of a semisimple ideal and an abelian center. In this setting, the decomposition exists for every element, with the center contributing entirely to the semisimple part, as central elements are semisimple (their adjoint action is zero, hence diagonalizable).^[14] Uniqueness holds analogously, since reductive algebras embed into \mathfrak{gl}(V) for some representation V, inheriting the decomposition from the general linear case while preserving the abstract properties.^[3] Claude Chevalley reformulated the decomposition in terms of polynomials, showing that the semisimple part s is a polynomial in x such that \mathrm{ad}_s is the unique diagonalizable operator sharing the same characteristic polynomial as \mathrm{ad}_x. This polynomial expression, constructed via interpolation (e.g., using the Chinese Remainder Theorem on the minimal and characteristic polynomials of \mathrm{ad}_x), ensures s captures the eigenvalues of \mathrm{ad}_x without the nilpotent contributions, providing an algebraic characterization independent of the representation.^[3]

Applications to representations

In the context of representation theory, the Jordan decomposition of elements in a Lie algebra \mathfrak{g} plays a crucial role in analyzing representations \rho: \mathfrak{g} \to \mathfrak{gl}(V) on finite-dimensional vector spaces V over an algebraically closed field of characteristic zero. For any x \in \mathfrak{g}, the unique decomposition x = x_s + x_n with x_s semisimple and x_n nilpotent induces \rho(x) = \rho(x_s) + \rho(x_n) in \mathfrak{gl}(V), where \rho(x_s) is semisimple and \rho(x_n) is nilpotent, preserving the commuting property. This structure facilitates the classification of representations by decomposing the action into diagonalizable (semisimple) and strictly triangular (nilpotent) components, enabling the study of weight spaces and highest weight modules in semisimple Lie algebras. For instance, in the representation theory of semisimple Lie algebras, the semisimple part determines the toral action, while the nilpotent part governs indecomposable extensions, as seen in the complete reducibility of finite-dimensional representations via Weyl's theorem.^[15]^[16] The Jordan decomposition also connects directly to the structure of Borel subgroups in the associated algebraic groups. In a semisimple algebraic group G with Lie algebra \mathfrak{g}, a Borel subgroup B decomposes as B = T \cdot U, where T is a maximal torus and U its unipotent radical; correspondingly, for x \in \mathfrak{b} (the Lie algebra of B), the semisimple part x_s lies in the Lie algebra of T, while the nilpotent part x_n lies in the Lie algebra of U. This alignment aids in understanding representations restricted to Borel subgroups, where the semisimple component fixes weight decompositions and the nilpotent component acts on flag modules. Such relations underpin the study of induced representations and parabolic induction in the representation theory of algebraic groups.^[15]^[17] A concrete example arises in \mathrm{GL}_n, where the Jordan decomposition classifies elements of \mathfrak{gl}_n up to conjugation, with semisimple elements conjugate to diagonal matrices and nilpotent elements to strictly upper triangular ones. This classification links to the geometry of flag varieties, as the centralizer of an element reflects stabilizers of flags under the group action, facilitating the decomposition of representations into those parameterized by partitions or Young diagrams. In broader terms, for semisimple groups, the decomposition induces compatibility with the Bruhat decomposition G = \bigcup_{w \in [W](/page/W)} B w B, where W is the Weyl group; semisimple elements centralize within tori, and their conjugation classes intersect Bruhat cells, enabling enumerative results in representation theory such as dimension formulas for induced modules.^[17]^[15]

In measure theory

Hahn-Jordan decomposition of signed measures

In measure theory, a signed measure on a measurable space (X, \Sigma) is a \sigma-additive set function \mu: \Sigma \to \mathbb{R} that satisfies \mu(\emptyset) = 0, allowing both positive and negative values on measurable sets.^[18] The Hahn-Jordan decomposition theorem asserts that any such signed measure \mu can be uniquely expressed as the difference of two positive measures \mu^+ and \mu^-, where \mu = \mu^+ - \mu^-, the supports of \mu^+ and \mu^- are disjoint (i.e., \mu^+ \perp \mu^-), and the total variation measure |\mu| = \mu^+ + \mu^- is a positive measure.^[19] This decomposition provides a canonical way to separate the "positive" and "negative" contributions of \mu.^[20] The construction relies on a Hahn decomposition, which partitions X into two measurable sets P and N (a positive set and a negative set, respectively) such that P \cup N = X, P \cap N = \emptyset, \mu(E \cap P) \geq 0 for all E \in \Sigma, and \mu(E \cap N) \leq 0 for all E \in \Sigma.^[18] Such a partition always exists by the Hahn decomposition theorem, and the positive and negative parts are then defined by \mu^+(E) = \mu(E \cap P) and \mu^-(E) = -\mu(E \cap N) for all E \in \Sigma; these are nonnegative measures because of the properties of P and N.^[21] The Hahn decomposition is unique up to sets of |\mu|-measure zero, which implies that the Jordan decomposition \mu = \mu^+ - \mu^- is also unique up to |\mu|-null sets.^[22] A concrete example arises in the context of Lebesgue-Stieltjes measures generated by functions of bounded variation. Consider the step function F: \mathbb{R} \to \mathbb{R} defined by F(x) = 0 for x < 0, F(x) = 1 for $0 \leq x < 1, and F(x) = 0 for x \geq 1; this induces a signed Lebesgue-Stieltjes measure \mu on the Borel \sigma-algebra of \mathbb{R} with point masses \mu(\{0\}) = 1 and \mu(\{1\}) = -1, and \mu(E) = 0 for any Borel set E disjoint from \{0, 1\}.^[23] The Hahn decomposition takes P = \{0\} (or more generally, sets containing 0 but not 1) and N = \{1\} (or sets containing 1 but not 0), yielding \mu^+(E) = \mu(E \cap \{0\}) = \delta_0(E) (the Dirac measure at 0) and \mu^-(E) = -\mu(E \cap \{1\}) = \delta_1(E) (the Dirac measure at 1), with disjoint supports.^[24]

Properties and uniqueness

The total variation of a signed measure \mu on a measurable set E is defined as

|\mu|(E) = \sup\left\{ \sum_i |\mu(E_i)| : \{E_i\} \text{ is a finite partition of } E \right\},

where the supremum is taken over all finite partitions of E into measurable sets.^[25] This total variation measure |\mu| coincides with \mu^+ + \mu^- in the Jordan decomposition \mu = \mu^+ - \mu^-, where \mu^+ and \mu^- are mutually singular non-negative measures.^[21] A signed measure \mu is positive if and only if \mu^- = 0, meaning \mu assigns non-negative values to all measurable sets.^[25] The Jordan decomposition is minimal among all possible decompositions into differences of non-negative measures. Specifically, if \mu = \nu^+ - \nu^- is another decomposition into positive measures, then \nu^+ \geq \mu^+ and \nu^- \geq \mu^- on every measurable set, with equality holding almost everywhere due to mutual singularity when \nu^+ \perp \nu^-.^[22] This minimality implies that the supports of \mu^+ and \mu^- cannot be refined further into coarser disjoint sets while preserving the decomposition, as any alternative would not be strictly positive and negative on those supports.^[22] The Jordan decomposition exists for any signed measure \mu on a measurable space. To prove existence, consider the collection of all positive sets for \mu; by Zorn's lemma applied to this partially ordered family under inclusion, there exists a maximal positive set P, and the complement N = X \setminus P is then a maximal negative set, forming a Hahn decomposition \{P, N\}.^[25] The positive and negative parts are then defined as \mu^+(E) = \mu(E \cap P) and \mu^-(E) = -\mu(E \cap N), yielding the Jordan decomposition.^[25] Uniqueness follows from the essential uniqueness of the Hahn decomposition. If \{P, N\} and \{Q, R\} are two Hahn decompositions of \mu, then |\mu|(P \triangle Q) = 0. Thus, for any measurable E, \mu(E \cap P) = \mu(E \cap Q) almost everywhere with respect to |\mu|, implying \mu^+ = \nu^+ and \mu^- = \nu^- almost everywhere with respect to |\mu|.^[25]^[22] The decomposition holds for arbitrary signed measures \mu. In the case where |\mu| is not \sigma-finite, care must be taken with the supports, as the positive and negative sets may have infinite measure, but uniqueness still holds up to null sets of |\mu|.^[21]

Relation to Lebesgue decomposition

The Lebesgue decomposition theorem asserts that given a σ-finite positive measure λ on a measurable space and a signed measure μ, there exist unique signed measures μ_ac and μ_s such that μ = μ_ac + μ_s, where μ_ac is absolutely continuous with respect to λ (μ_ac ≪ λ) and μ_s is singular with respect to λ (μ_s ⊥ λ).^[26] Each of these signed measures μ_ac and μ_s admits its own Hahn-Jordan decomposition into positive and negative parts, extending the structure provided by the Jordan decomposition to components defined relative to the reference measure λ. The Jordan decomposition integrates with the Lebesgue decomposition through the total variation measure: if |μ| denotes the total variation of μ arising from its Hahn-Jordan decomposition, then |μ| = |μ_ac| + |μ_s|, reflecting the mutual singularity of the absolutely continuous and singular components.^[26] Furthermore, applying the Hahn-Jordan decomposition to μ_ac yields positive measures whose Radon-Nikodym derivatives with respect to λ provide the density function for the absolutely continuous part, as guaranteed by the Radon-Nikodym theorem. In probability theory, this combined framework decomposes signed measures on probability spaces into parts amenable to martingale analysis, such as in Doob's decomposition of submartingales into a martingale and an increasing predictable process, where the Lebesgue singular component captures discontinuities or jumps orthogonal to the reference filtration. The uniqueness of both the Lebesgue and Hahn-Jordan decompositions ensures that these representations are canonical, facilitating convergence results and conditional expectation computations in stochastic processes. For a concrete illustration, consider the signed measure μ defined as the difference between Lebesgue measure on [0,1] and the Dirac measure at 0.5; the Lebesgue decomposition with respect to Lebesgue measure λ yields μ_ac as the Lebesgue measure itself (absolutely continuous) and μ_s as the negative Dirac measure (singular, concentrated at {0.5}), with the Jordan decompositions of each part aligning with their supports—μ_ac^+ coinciding with λ and μ_ac^- = 0 on [0,1], while μ_s^+ = 0 and μ_s^- is the Dirac measure.

References

[1]
[PDF] Notes on Jordan Decomposition - LSU Math
Theorem 1 (Jordan Normal Form). Let V be a finite-dimensional vector space. For any linear operator g ∈ End(V ), there exists an element x ∈ GL(V ) such ...Missing: algebra | Show results with:algebra
[2]
[PDF] Lecture 4 - The Fitting and Jordan-Chevalley decompositions
Sep 18, 2012 · If x : V → V is an endomorphism, xs is called its semisimple part and xn is called its nilpotent part. Proposition 2.4 Given an endomorphism x : ...
[3]
[PDF] The Jordan-Chevalley decomposition - The University of Chicago
Sep 8, 2014 · This paper illustrates the Jordan-Chevalley decomposition through two related problems. Contents. 1. Introduction. 1. 2. Linear Algebra Review.
[4]
[PDF] SEMISIMPLICITY A subspace W of an F-vector space V always has ...
The operator A: V → V is semisimple if and only if its minimal polynomial ... A polynomial which splits with distinct roots is separable, and a polynomial which ...
[5]
[PDF] 4. Jordan decomposition and Cartan's criterion - Brandeis
Sep 20, 2011 · Jordan decomposition. The proof requires the ground field F to ... semisimple, (ad x)n is nilpotent and they commute. So, the lemma is ...
[6]
Traité des substitutions et des équations algébriques - Internet Archive
Dec 3, 2008 · Traité des substitutions et des équations algébriques. by: Jordan, Camille, 1838-1922. Publication date: 1870. Topics: Galois theory, Groups ...
[7]
[PDF] 3 Jordan Canonical Forms - UC Berkeley math
The block-diagonal matrices described in the theorem are called Jordan canonical forms (or Jordan normal forms). Their diag- onal blocks are called Jordan ...
[8]
Jordan Block -- from Wolfram MathWorld
A Jordan canonical form consists of one or more Jordan blocks. The convention that 1s be along the subdiagonal instead of the superdiagonal is sometimes adopted ...Missing: structure | Show results with:structure
[9]
[PDF] How to find the Jordan canonical form of a matrix
Apr 10, 2013 · To find the Jordan canonical form, first find all eigenvectors, then the number of linearly independent eigenvectors determines the number of ...
[10]
[PDF] Homework 6
Using Jordan canonical form and generalized eigenspaces, prove that there is a unique expression. T = Tss + Tn where Tss and Tn are a pair of commuting ...Missing: normal | Show results with:normal
[11]
[PDF] Lecture 12 — Structure Theory of Semisimple Lie Algebras (I)
Oct 19, 2010 · ad an is nilpotent, and ad as is semisimple, we conclude that a = a0 ... Hence we check that a = Φas + Φan is an abstract Jordan decomposition of ...
[12]
https://arxiv.org/pdf/1906.01918.pdf
[13]
[PDF] part i: geometry of semisimple lie algebras
that for every x ∈ g there exists a unique decomposition x = xss + xn into a semisimple part xss and a nilpotent part xn such that [xss,xn] = 0. Uniqueness ...
[14]
https://arxiv.org/pdf/1008.1217.pdf
[15]
[PDF] Lie Algebras, Algebraic Groups, and Lie Groups - James Milne
May 5, 2013 · These notes are an introduction to Lie algebras, algebraic groups, and Lie groups in characteristic zero, emphasizing the relationships between ...
[16]
[PDF] Introduction to Lie Algebras and Representation Theory
(1) The Jordan-Chevalley decomposition of linear transformations is emphasized, with "toral" subalgebras replacing the more traditional Cartan subalgebras ...
[17]
None
Below is a merged summary of Jordan Decomposition and related topics in T.A. Springer’s *Linear Algebraic Groups* (1998), consolidating all information from the provided segments into a comprehensive response. To maximize detail and clarity, I will use a structured format with tables where appropriate, followed by a narrative summary for additional context. The response retains all key points, theorems, page references, and URLs mentioned across the segments.
[18]
[PDF] Differentiation Lecture 7, Following Folland, ch 3.1, 3.2
1.2. Hahn decomposition. We will show that any signed measure is a difference of two measures (as in Example 1.1 above). Definition 1.2 ...
[19]
[PDF] Differentiation - UC Davis Mathematics
To prove the Jordan decomposition of a signed measure, we first show that a measure space can be decomposed into disjoint subsets on which a signed measure is ...
[20]
[PDF] 17.2. Signed Measures: Hahn and Jordan Decompositions—Proofs
Apr 19, 2017 · The Jordan Decomposition Theorem. Let ν be a signed measure on (X,M). Then there are two mutually singular measures ν+ and ν− on ...
[21]
[PDF] 7. Signed measures and complex measures - KSU Math
The Hahn-Jordan decomposition has the following interesting application to the properties of the natural order relation on “honest” measures. The result below.
[22]
[PDF] 13. Complex Measures, Radon-Nikodym Theorem and the Dual of Lp
Jordan Decomposition. Definition 13.20. Let X = P ∪ N be a Hahn decomposition of ν and define ν+(E) = ν(P ...
[23]
[PDF] Measure Theory John K. Hunter - UC Davis Math
The following examples illustrate the three basic types of Lebesgue-Stieltjes measures. Example 2.35. If F(x) = x, then µF is Lebesgue measure on R with. µF ((a ...
[24]
[PDF] 5 Measure theory II - UCSB Math
The Hahn decomposition of Ω for a charge φ leads to the Jordan decom- position of φ into the positive and negative parts.
[25]
[PDF] Signed Measures
A pair {P, N} of elements in A for which P is positive N is negative, P ∪ N = X and P ∩ N = ∅ is called a Hahn decomposition of X with respect to µ. Remark 4.2.
[26]
[PDF] Existence, uniqueness, and minimality of the Jordan measure ... - arXiv
Jun 27, 2012 · Every finite signed measure µ has a unique decomposition into a difference µ = µ+ − µ− of two non-negative finite measures µ+ and. µ− such that ...
[27]
245B, notes 1: Signed measures and the Radon-Nikodym ...
Jan 4, 2009 · In this section we investigate the structure of this space, together with the closely related spaces of signed measures and finite measures.