Jordan Decomposition Theorem Research Articles

A geometric Jordan decomposition theorem

A geometric Jordan decomposition theorem

Higher order derivatives of the adjugate matrix and the Jordan form

In this short note, we show that the higher-order derivatives of the adjugate matrix Adj(z−A), are related to the nilpotent matrices and projections in the Jordan decomposition of the matrix A. These relations appear as a factorization of the derivative of the adjugate matrix as a product of factors related to the eigenvalues, nilpotent matrices and projectors. The novel relations are obtained using the Riesz projector and functional calculus. The results presented here can be considered to be a generalization of Thompson and McEnteggert's theorem relating the adjugate matrix to the orthogonal projection on the eigenspace of simple eigenvalues for symmetric matrices. They can also be seen as a complement to some earlier results by B. Parisse, M. Vaughan that relate derivatives of the adjugate matrix to the invariant subspaces associated with an eigenvalue. Our results can also be interpreted as a general eigenvector-eigenvalue identity. Many previous works have dealt with relations between the projectors on the eigenspaces and the derivatives of the adjugate matrix with the characteristic spaces but it seems that there is no explicit mention in the literature of the factorization of the higher-order derivatives of the adjugate matrix as a matrix multiplication involving nilpotent and projector matrices, which appear in the Jordan decomposition theorem.

Remarks on the effective Jordan decomposition

According to the Jordan Decomposition Theorem, for any function f:[0,1]→R of bounded variation there exists a pair of nondecreasing functions f1,f2:[0,1]→R with f=f1−f2. Such a pair is called a Jordan decomposition of f. Ko (1991) and Zheng and Rettinger (2005) have analyzed computability-theoretic and complexity-theoretic aspects of this theorem. We complement their observations by three new results. One of our results says that there exists a polynomial time computable, absolutely continuous function f:[0,1]→R with polynomial modulus of absolute continuity such that, on the one hand, the so-called standard Jordan decomposition of f is computable but, on the other hand, neither the standard Jordan decomposition of f nor any other Jordan decomposition of f can be computed easily, say, in polynomial time, or in exponential time.

Free noncommutative hereditary kernels: Jordan decomposition, Arveson extension, kernel domination

We discuss (i) a quantized version of the Jordan decomposition theorem for a complex Borel measure on a compact Hausdorff space, namely, the more general problem of decomposing a general noncommutative kernel (a quantization of the standard notion of kernel function) as a linear combination of completely positive noncommutative kernels (a quantization of the standard notion of positive definite kernel). Other special cases of (i) include: the problem of decomposing a general operator-valued kernel function as a linear combination of positive kernels (not always possible), of decomposing a general bounded linear Hilbert-space operator as a linear combination of positive linear operators (always possible), of decomposing a completely bounded linear map from a C^* -algebra \mathcal{A} to an injective C^* -algebra \mathcal{L}(\mathcal{Y}) as a linear combination of completely positive maps from \mathcal{A} to \mathcal{L}(\mathcal{Y}) (always possible). We also discuss (ii) a noncommutative kernel generalization of the Arveson extension theorem (any completely positive map \phi from an operator system \mathbb{S} to an injective C^* -algebra \mathcal{L}(\mathcal{Y}) can be extended to a completely positive map \phi_e from a C^* -algebra containing \mathbb{S} to \mathcal{L}(\mathcal{Y})) , and (iii) a noncommutative kernel version of a Positivstellensatz (i.e., finding a certificate to explain why one kernel is positive at points where another given kernel is strictly positive).

THE REVERSE MATHEMATICS OF THEOREMS OF JORDAN AND LEBESGUE

AbstractThe Jordan decomposition theorem states that every function $f \colon \, [0,1] \to \mathbb {R}$ of bounded variation can be written as the difference of two non-decreasing functions. Combining this fact with a result of Lebesgue, every function of bounded variation is differentiable almost everywhere in the sense of Lebesgue measure. We analyze the strength of these theorems in the setting of reverse mathematics. Over $\mathsf {RCA}_{0}$ , a stronger version of Jordan’s result where all functions are continuous is equivalent to $\mathsf {ACA}_0$ , while the version stated is equivalent to ${\textsf {WKL}}_{0}$ . The result that every function on $[0,1]$ of bounded variation is almost everywhere differentiable is equivalent to ${\textsf {WWKL}}_{0}$ . To state this equivalence in a meaningful way, we develop a theory of Martin–Löf randomness over $\mathsf {RCA}_0$ .

On approximately monotone and approximately H\xf6lder functions

A real valued function f defined on a real open interval I is called Phi -monotone if, for all x,yin I with xle y it satisfies f(x)≤f(y)+Φ(y-x),\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} f(x)\\le f(y)+\\Phi (y-x), \\end{aligned}$$\\end{document}where Phi :[0,ell (I) [ rightarrow mathbb {R}_+ is a given nonnegative error function, where ell (I) denotes the length of the interval I. If f and -f are simultaneously Phi -monotone, then f is said to be a Phi -Hölder function. In the main results of the paper, we describe structural properties of these function classes, determine the error function which is the most optimal one. We show that optimal error functions for Phi -monotonicity and Phi -Hölder property must be subadditive and absolutely subadditive, respectively. Then we offer a precise formula for the lower and upper Phi -monotone and Phi -Hölder envelopes. We also introduce a generalization of the classical notion of total variation and we prove an extension of the Jordan Decomposition Theorem known for functions of bounded total variations.

Solutions and connections of nonlocal derivative nonlinear Schrödinger equations

All possible nonlocal versions of the derivative nonlinear Schrodinger equations are derived by the nonlocal reduction from the Chen–Lee–Liu equation, the Kaup–Newell equation and the Gerdjikov–Ivanov equation which are gauge equivalent to each other. Their solutions are obtained by composing constraint conditions on the double Wronskian solution of the Chen–Lee–Liu equation and the nonlocal analogues of the gauge transformations among them. Through the Jordan decomposition theorem, those solutions of the reduced equations from the Chen–Lee–Liu equation can be written as canonical form within real field.

Signed $$\lambda $$λ-Measures on Effect Algebras

The concepts of \(\lambda \)-measures and signed \(\lambda \)-measures (nonadditive measures) defined on effect algebras are introduced and investigated in this paper. A few examples of \(\lambda \)-measures and signed \(\lambda \)-measures are also given. Moreover, the Jordan decomposition theorem for signed \(\lambda \)-measures defined on effect algebras is also presented.

Differences of two semiconvex functions on the real line

It is proved that real functions on $\mathbb R$ which can be represented as the difference of two semiconvex functions with a general modulus (or of two lower $C^1$-functions, or of two strongly paraconvex functions) coincide with semismooth functions on $\mathbb R$ (i.e. those locally Lipschitz functions on $\mathbb R$ for which $f'_+(x) = \lim_{t \to x+} f'_+(t)$ and $f'_-(x) = \lim_{t \to x-} f'_-(t)$ for each $x$). Further, for each modulus $\omega$, we characterize the class $DSC_{\omega}$ of functions on $\mathbb R$ which can be written as $f=g-h$, where $g$ and $h$ are semiconvex with modulus $C\omega$ (for some $C>0$) using a new notion of $[\omega]$-variation. We prove that $f \in DSC_{\omega}$ if and only if $f$ is continuous and there exists $D>0$ such that $f'_+$ has locally finite $[D \omega]$-variation. This result is proved via a generalization of the classical Jordan decomposition theorem which characterizes the differences of two $\omega$-nondecreasing functions (defined by the inequality $f(y) \geq f(x)- \omega(y-x)$ for $y>x$) on $[a,b]$ as functions with finite $[2\omega]$-variation. The research was motivated by a recent article by J. Duda and L. Zajíček on Gâteaux differentiability of semiconvex functions, in which surfaces described by differences of two semiconvex functions naturally appear.

Linear Algebra via Complex Analysis

The resolvent (λI − A)−1 of a matrix A is naturally an analytic function of λ ∈ ℂ, and the eigenvalues are isolated singularities. We compute the Laurent expansion of the resolvent about the eigenvalues of A. Using the Laurent expansion, we prove the Jordan decomposition theorem, prove the Cayley–Hamilton theorem, and determine the minimal polynomial of A. The proofs do not make use of determinants, and many results naturally generalize to operators on Banach spaces.

On statistical measure theory

The purpose of this paper is to unify various kinds of statistical convergence by statistical measure convergence and to present Jordan decomposition of finitely additive measures. It is done through dealing with the most generalized statistical convergence–ideal convergence by applying geometric functional analysis and Banach space theory. We first show that for each type of ideal I(⊂2N) convergence, there exists a set S of statistical measures such that the measure S-convergence is equivalent to the statistical convergence. To search for Jordan decomposition of measures of statistical type, we show that the subspace XI≡span¯{χA:A∈I} is an ideal of the space ℓ∞ in the sense of Banach lattice, hence the quotient space ℓ∞/XI is isometric to a C(K) space. We then prove that a statistical measure has a Jordan decomposition if and only if its corresponding functional is norm-attaining on ℓ∞, and which in turn induces an approximate null–ideal preserved Jordan decomposition theorem of finitely additive measures. Finally, we show this characterization and the approximate decomposition theorem are true for finitely additive measures defined on a general measurable space.

DECOMPOSITIONS OF SPACES OF MEASURES

Let 𝔐 be the Banach space of σ-additive complex-valued measures on an abstract measurable space. We prove that any closed, with respect to absolute continuity norm-closed, linear subspace L of 𝔐 is complemented and describe the unique complement, projection onto L along which has norm 1. Using this fact we prove a decomposition theorem, which includes the Jordan decomposition theorem, the generalized Radon–Nikodým theorem and the decomposition of measures into decaying and non-decaying components as particular cases. We also prove an analog of the Jessen–Wintner purity theorem for our decompositions.

Some properties of the variations of non-additive set functions on T-tribes

The inclusion variation and chain variation on T ∞ -tribes play an important role in T ∞ -measures and fuzzy games theory, where they were used for additive set functions on T ∞ -tribe. In this paper classical variation theory is extended to non-additive set functions on T-tribes, after defining the inclusion variation, disjoint and chain variation on T-tribes, we investigate some properties of these three variations in detail, such as, the (null-) null-additivity, exhaustivity, order continuity, continuity from the left and so on. An alternative proof of the Jordan decomposition theorem is given. Thus, some results pertaining to variations in classical measure theory are generalized. In some case, as for T ∞ -measures on T ∞ -tribe, its different formal variations can be extended to T s -measures on T s -tribes.

Decision Effects, the Hahn–Jordan Decomposition for States, and Their Connection to Ludwig’s Axiomatic Approach to Quantum Mechanics

We begin with a quantum logic carrying a large collection of states. We then form a dual pair of Banach spaces—base normed and order unit normed—containing the states and the logic, respectively. A Galois connection on the face lattices of the states and the dual positive order unit interval is introduced. The elements of the logic are connected to a dense subset of the extreme points of this order interval in the order unit space using a generalized form of the Hahn–Jordan decomposition theorem. Decision effects are defined and identified with the elements of the original logic. Finally, an important axiom of Ludwig is introduced which ties together all the lattices of Galois closed faces of states, Galois closed order intervals of the positive order unit interval, decision effects, and the original quantum logic. The emphasis here is on the consequences of functional analytic assumptions. The paper concludes with a simple example where Ludwig’s axiom does not hold and we see parts of the theory dissolve.

Effective Jordan Decomposition

The Jordan Decomposition Theorem says that any real function of bounded variation can be expressed as a difference of two increasing functions. This paper investigates the effective version of Jordan decomposition and discusses the properties of variation of computable real functions. First we show that the effective version of Jordan decomposition does not hold in general. Then we give a sufficient and necessary condition for computable real functions of bounded variation which have an effective Jordan decomposition. Applying this condition, we construct a computable real function which has a computable modulus of absolute continuity (hence of bounded variation) but is not effectively Jordan decomposable. Finally, we prove a version of this result restricted to polynomial time which answers negatively an open question of Ko in [8].

Decay measures on locally compact abelian topological groups

We show that the Banach space M of regular σ-additive finite Borel complex-valued measures on a non-discrete locally compact Hausdorff topological Abelian group is the direct sum of two linear closed subspaces MD and MND, where MD is the set of measures μ ∈ M whose Fourier transform vanishes at infinity and MND is the set of measures μ ∈ M such that ν ∉ MD for any ν ∈ M {0} absolutely continuous with respect to the variation |μ|. For any corresponding decomposition μ = μD + μND (μD ∈ MD and μND ∈ MND) there exist a Borel set A = A(μ) such that μD is the restriction of μ to A, therefore the measures μD and μND are singular with respect to each other. The measures μD and μND are real if μ is real and positive if μ is positive. In the case of singular continuous measures we have a refinement of Jordan's decomposition theorem. We provide series of examples of different behaviour of convolutions of measures from MD and MND.

Some properties of the variations of non-additive set functions I

The disjoint variation and chain variation in classical measure theory play an important role in the decompositions of signed measures, where they coincide. But they are usually different for non-additive set functions. In this paper, we discuss some properties of the variations, such as, the ( null -) null -additivity, exhaustivity, order continuity, continuity and so on. A version of the Jordan decomposition theorem is proved for signed lower semicontinuous fuzzy measures.

A short proof of the jordan decomposition theorem

We present a short proof of the Jordan Decomposition Theorem

On the characterization of exponential polynomials

Exponential polynomials, i.e. solutions of linear homogeneous differential equations with constant coefficients, are characterized as continuous solutions of linear homogeneous difference equations with constant coefficients. Namely, denote by C the \({\Bbb C}\)-linear space of all continuous functions on \({\Bbb R}\), and define for \( \omega \in {\Bbb R}_+\) the shift operator \(z_{\omega }:C\to C\) by \(\eta \mapsto z_{\omega }\eta \) such that \(z_{\omega }\eta (t)=\eta (t+{\omega })\) for all \(t\in {\Bbb R}\). Every ideal \({\mathfrak a}\) of \(R:={\Bbb C}[x,y]\) determines a subspace¶¶\( C_{\frak a}=\{\eta \in C:f(z_{{\omega }_1},z_{{\omega }_2})\eta =0\) for all \(f(x,y)\in {\frak a}\}\,.\)¶¶For incommensurable \({\omega }_1,{\omega }_2\in {\Bbb R}_+\) and ideals \({\frak a}\) in R with a factor ring \(R/{\frak a}\) , which is finite dimensional as a \({\Bbb C}\)-linear space, the set \(C_{\mathfrak a}\) is characterized. The main idea consists in applying the Kronecker approximation theorem to the analytic transition from the explicit representation of the solutions of recurrence sequences to that of the continuous solutions of difference equations. For an alternative proof, the Kronecker approximation theorem is combined with the Jordan decomposition theorem.

Belief functions generated by signed measures

It is a well-known fact that the usual and already classical combinatorial definition of belief function over (the powerset of) a finite set can be generalized in such a way that belief function is defined by the quantile function of a set-valued (generalized) random variable defined over an abstract probability space. In this contribution we shall investigate a further stage of generalization resulting when the probability space in question is replaced by a measurable space equipped by a signed measure; signed measure is a σ-additive set function which can take values also outside the unit interval, including the negative and infinite ones. An assertion analogous to the Jordan decomposition theorem for signed measures is stated and proved, according to which each signed belief function restricted to its finite values can be defined by a linear combination of two classical probabilistic belief functions, supposing that the basic set is finite.