Common Eigenvector Research Articles

Solving Laguerre–Gauss Mode Eigenvectors and Eigenfunctions inEntangled State Representation

We solve the Laguerre–Gauss mode eigenvectors and eigenfunctions in theentangled state representation by searching for common eigenvectors ofthe 2-dimensional harmonic oscillator's total energy operator and theangular momentum operator. We find that in the entangled staterepresentation the eigen-solution satisfies the Hukuhara equation, and itssolution is confluent hypergeometric function.

State-Vector Space and Canonical Coherent States in NoncommutativePlane

The structure of the state-vector space of identical bosons innoncommutative spaces is investigated. To maintain Bose–Einsteinstatistics the commutation relations of phase space variablesshould simultaneously include coordinate-coordinatenon-commutativity and momentum-momentum non-commutativity, whichleads to a kind of deformed Heisenberg-Weyl algebra. Although thereis no ordinary number representation in this state-vector space,several set of orthogonal and complete state-vectors can bederived which are common eigenvectors of corresponding pairs ofcommuting Hermitian operators. As a simple application of thisstate-vector space, an explicit form of two-dimensional canonicalcoherent state is constructed and its properties are discussed.

Sylvester-’t Hooft generators and relations between them for $$\mathfrak{s}\mathfrak{l}(n)$$ and $$\mathfrak{g}\mathfrak{l}(n|n)$$

Among the simple finite-dimensional Lie algebras, only \(\mathfrak{s}\mathfrak{l}(n)\) has two finite-order automorphisms that have no common nonzero eigenvector with the eigenvalue one. It turns out that these automorphisms are inner and form a pair of generators that allow generating all of \(\mathfrak{s}\mathfrak{l}(n)\) under bracketing. It seems that Sylvester was the first to mention these generators, but he used them as generators of the associative algebra of all n×n matrices Mat(n). These generators appear in the description of elliptic solutions of the classical Yang-Baxter equation, the orthogonal decompositions of Lie algebras, ’t Hooft’s work on confinement operators in QCD, and various other instances. Here, we give an algorithm that both generates \(\mathfrak{s}\mathfrak{l}(n)\) and explicitly describes a set of defining relations. For simple (up to the center) Lie superalgebras, analogues of Sylvester generators exist only for \(\mathfrak{g}\mathfrak{l}(n|n)\). We also compute the relations for this case.

Commuting and stable feedback design for switched linear systems

In the paper, commuting and stable feedback design for switched linear systems is investigated. This problem is formulated as to build up suitable state feedback controller for each subsystem such that the closed-loop systems are not only asymptotically stable but also commuting each other. A new concept, common admissible eigenvector set (CAES), is introduced to establish necessary/sufficient conditions for commuting and stable feedback controllers. For second-order systems, a necessary and sufficient condition is established. Moreover, a parametrization of the CAES is also obtained. The motivation comes from stabilization of switched linear systems which consist of a family of LTI systems and a switching law specifying the switching between them, where if all the subsystems are stable and commuting each other, then the total system is stable under arbitrary switching.

Application of new entangled state representation to a two-coupled oscillator system

We study the eigenstate problem of a two-coupled oscillator system. A new entangled state representation |γ composed of the common eigenvectors of operators (x1+p2) and (p1+x2) is established. Eigenvalues and eigenvectors of the Hamiltonian are obtained in |γ representations. The same problem is studied in the second-quantization representation. We find that the second-quantization representation can be used to derive the normally ordered product expression of eigenvector in Fock space. In particular, we find that the ground state of the Hamiltonian is a kind of generalized two-mode squeezed state.

Common fixed points and common eigenvectors for sets of matrices

The following questions are studied: Under what conditions does the existence of a (nonzero) fixed point for every member of a semigroup of matrices imply a common fixed point for the entire semigroup? What is the smallest number k such that the existence of a common fixed point for every k members of a semigroup implies the same for the semigroup? If every member has a fixed space of dimension at least k: What is the best that can be said about the common fixed space? We also consider analogs of these questions with general eigenspaces replacing fixed spaces.

A Theorem of Krein Revisited

M. Krein proved in [KR48] that if T is a continuous operator on a normed space leaving invariant an open cone, then its adjoint T ∗ has an eigenvector. We present generalizations of this result as well as some applications to C∗-algebras, operators on `1, operators with invariant sets, contractions on Banach lattices, the Invariant Subspace Problem, and von Neumann algebras. M. Krein proved in [KR48, Theorem 3.3] that if T is a continuous operator on a normed space leaving invariant a non-empty open cone, then its adjoint T ∗ has an eigenvector. Krein’s result has an immediate application to the Invariant Subspace Problem because of the following observation. If T is a bounded operator on a Banach space and not a multiple of the identity, and T ∗f = λf , then the kernel of f is a closed non-trivial subspace of codimension 1 which is invariant under T . Moreover, Range(λI − T ) is a closed nontrivial subspace which is proper (it is contained in the kernel of f) and hyperinvariant for T , that is, it is invariant under every operator commuting with T . Several proofs and modifications of Krein’s theorem appear in the literature, see, e.g., [AAB92, Theorems 6.3 and 7.1] and [S99, p. 315]. We prove yet another version of Krein’s Theorem: if T is a positive operator on an ordered normed space in which the unit ball has a dominating point, then T ∗ has a positive eigenvector. We deduce the original Krein’s version of the theorem from this, as well as several applications and related results. In particular, we show that if a bounded operator T on a Banach space satisfies any of the following conditions, then T ∗ has an eigenvector. Moreover, if the condition holds for a commutative family of operators, then the family of the adjoint operators has a common eigenvector. • T leaves invariant a cone with an interior point; • T is a positive operator on a unital C∗-algebra; 2000 Mathematics Subject Classification. Primary: 46B40, 47B60, 47B65; Secondary: 47A15, 47B48, 46L05, 46L10.

Learning a multivariate Gaussian mixture model with the reversible jump MCMC algorithm

This paper is a contribution to the methodology of fully Bayesian inference in a multivariate Gaussian mixture model using the reversible jump Markov chain Monte Carlo algorithm. To follow the constraints of preserving the first two moments before and after the split or combine moves, we concentrate on a simplified multivariate Gaussian mixture model, in which the covariance matrices of all components share a common eigenvector matrix. We then propose an approach to the construction of the reversible jump Markov chain Monte Carlo algorithm for this model. Experimental results on several data sets demonstrate the efficacy of our algorithm.

EINSTEIN–PODOLSKY–ROSEN ENTANGLED STATE REPRESENTATION FOR ANGULAR-MOMENTUM AND RADIUS ENTANGLEMENT

By comparison with the Einstein–Podolsky–Rosen coordinate-momentum entangled state, which is the common eigenvector of two particles' relative coordinate and total momentum, we construct a new quantum mechanical entangled state representation, namely, the entangled state representation of angular-momentum and radius. A concrete physical system which can embody the new quantum entanglement is analyzed.

Entangled State Representation for Hamiltonian Operator of Quantum Pendulum**The project supported by the President Foundation of the Chinese Academy of Sciences

By virtue of the Einstein–Podolsky–Rosen entangled state, which is the common eigenvector of two particles' relative coordinate and total momentum, we establish the bosonic operator version of the Hamiltonian for a quantum point-mass pendulum. The Hamiltonian displays the correct Schrödinger equation in the entangled state representation. The corresponding Heisenberg operator equations which predict the angular momentum–angle uncertainty relation are derived. The quantum operator description of two quantum pendulums coupled by a spring is also derived.

The Fundamental Theorem of Algebra and Linear Algebra

The first widely accepted proof of the Fundamental Theorem of Algebra was published by Gaus in 1799 in his Ph.D. thesis, although to current standards this proof has gaps. Argand gave a proof (with only small gaps) in 1814 which was based on a flawed proof of d’Alembert of 1746. Many more proofs followed, including three more proofs by Gaus. For a more about the history of the Fundamental Theorem of Algebra, see [5, 6]. Proofs roughly can be divided up in three categories (see [3] for a collection of proofs). First there are the topological proofs (see [1, 8]). These proofs are based on topological considerations such as the winding number of a curve in R around 0. Gaus’ original proof might fit in this category as well. Then there are analytical proofs (see [9]) which are related to Liouville’s result that an entire non-constant function on C is unbounded. Finally there are the algebraic proofs (see [4, 10]). These proofs only use the facts that every odd polynomial with real coefficients has a real root, and that every complex number has a square root. The deeper reasons why these proves work can be understood in terms of Galois Theory. For a linear algebra course, the Fundamental Theorem of Algebra is needed, so it is therefore desireable to have a proof of it in terms of linear algebra. In this paper we will prove that every square matrix with complex coefficients has an eigenvector. This is equivalent to the Fundamental Theorem of Algebra. In fact we will prove the slightly stronger result that any number of commuting square matrices with complex entries will have a common eigenvector. The proof is entirely within the framework of linear algebra, and unlike most other algebraic proves of the Fundamental Theorem of Algebra, it does not require Galois Theory or splitting fields. Another (but longer) proof using linear algebra can be found in [7].

TRIPARTITE ENTANGLED STATE REPRESENTATION AND ITS APPLICATION IN QUANTUM TELEPORTATION

We set up a tripartite entangled state representation |p, χ2, χ3> in three-mode Fock space which is composed of the common eigenvectors of three particles' relative coordinates X1 - X2 and X1 - X3 as well as the total momentum P1 + P2 + P3. The Schmidt decomposition of |p, χ2, χ3 > is made and its application in quantum teleporting a two-particle entangled state or a two-mode squeezed state is analyzed.

Bethe ansatz for the Gaudin model and its relation with Knizhnik–Zamolodchikov equations

We recall the construction of the common eigenvectors of Gaudin Hamiltonians based on the Bethe ansatz. In the case of an arbitrary Lie algebra, this construction can be done either recursively or explicitly and we prove the equivalence of the two methods. We also prove that Bethe vectors are singular only if the Bethe equations are satisfied. In each eigenspace of the spin operator we construct additional common eigenvectors, having the same eigenvalue as the vacuum vector and which can not be obtained by the Bethe ansatz. These eigenvectors are not singular. We also recall the connection between Bethe vectors and integral solutions of the KZ equation. In an analogous way, the additional vectors lead to solutions of KZ equation which are not singular vectors and do not have an integral representation.

New Three-Mode Einstein-Podolsky-Rosen Entangled StateRepresentation and Its Application in Squeezing Theory

By extending the Einstein-Podolsky-Rosen bipartite entanglement tothe tripartite case, we construct the common eigenvector of the tripartitecentre-of-mass coordinate and two mass-weighted relative momenta, whichis a new entangled state of continuum variables. The classical dilationtransform of variables in such a state induces a new three-modesqueezing operator related to a three-mode bosonic operator realizationof SU(1,1) Lie algebra.

Common Correlation Structures for Calibrating the LIBOR Model

In 1997 three papers that introduced very similar lognormal diffusion processes for interest rates appeared virtually simultaneously. These models, now commonly called the 'LIBOR models' are based on either lognormal diffusions of forward rates as in Brace, Gatarek & Musiela (1997) and Miltersen, Sandermann & Sondermann (1997) or lognormal diffusions of swap rates, as in Jamshidian (1997). The consequent research interest in the calibration of the LIBOR models has engendered a growing empirical literature, including many papers by Brigo and Mercurio, and Riccardo Rebonato. The art of model calibration requires a reasonable knowledge of option pricing and a thorough background in statistics - techniques that are quite different to those required to design no-arbitrage pricing models. Researchers will find the book by Brigo and Mercurio (2001) and the forthcoming book by Rebonato (2002) invaluable aids to their understanding. I aim to provide an accessible account of some interesting problems in LIBOR model calibration, but the ideas are complex, so notational complexities are reduced to a minimum. We take fresh look at three important modelling decisions when calibrating the LIBOR model: the parameterization of the correlation matrix for semi-annual forward rates; the use of principal component analysis in the orthogonal transform of the log normal forward rate model; and the iterations for recovering caplet volatilities from 'flat' cap volatilities. The first section provides the briefest of introductions to the lognormal forward rate version of the LIBOR model. Section two considers the calibration of the model to the cap market, where it is shown that the iteration of caplet volatilities from the 'flat' cap volatilities quoted in the market, should be performed by equating a vega weighted sum of caplet volatilities to the cap volatility. Section 3 discusses the more difficult problem of calibration to the swaption market. Two full rank parsimonious parameterizations of the semi-annual correlation matrix are specified, where correlations between annual forward rates are determined by the same parameters. Section 4 reconsiders calibration to the swaption market where the rank of forward rate correlation matrices is reduced by setting all but the three largest eigenvalues to zero. Rebonato (1999a), Rebonato and Joshi (2001) Hull and White (1999, 2000) and Longstaff, Santa-Clara and Schwartz (1999) have all found that this technique is useful for performing the simulations that are necessary for pricing path dependent options. The implication of zeroing eigenvalues is a transformation of the lognormal forward rate model where each forward rate is driven by three orthogonal factors that are derived from a principal component analysis. We show that, in fact, it is the common principal components model of Flury (1988) that should be used. That is, the same eigenvectors should be calibrated to all swaptions of the same tenor, and we advocate the use of market data rather than historical data for the calibration of these common eigenvectors.

A biplot method for multivariate normal populations with unequal covariance matrices

Some previous ideas about non-linear biplots to achieve a joint representation of multivariate normal populations and any parametric function without assumptions about the covariance matrices are extended. Usual restrictions on the covariance matrices (such as homogeneity) are avoided. Variables are represented as curves corresponding to the directions of maximum means variation. To demonstrate the versatility of the method, the representation of variances and covariances as an example of further possible interesting parametric functions have been developed. This method is illustrated with two different data sets, and these results are compared with those obtained using two other distances for the normal multivariate case: the Mahalanobis distance (assuming a common covariance matrix for all populations) and Rao’s distance, assuming a common eigenvector structure for all the covariance matrices.

Bipartite entangled state |$\zeta$$\rangle$ and its applications in quantum communication

We analyse the entanglement involved in the common eigenvector |ζ of the centre-of-mass coordinate and mass-weighted relative momentum of two particles. Its Schmidt decomposition exhibits that the |ζ state maximally entangles two single-mode coordinate (or momentum) eigenstates, of which one is squeezed. The squeezing parameter is related to the ratio of the masses of two particles. The |ζ state may describe the entangled state produced by a non-50/50 beamsplitter. Its applications in entanglement swapping and quantum dense coding are discussed. The corresponding entangling operators and swapping operators are derived.

Entanglement of the Common Eigenvector of Two Particles' Center-of-Mass Coordinate and Mass-Weighted Relative Momentum

We reveal that the common eigenvector of two particles' center-of-mass coordinate and mass-weighted relative momentum is an entangled state. Its Schmidt decomposition exhibits that the entanglement involves squeezing which depends on the ratio of two particles' masses. The corresponding entangling operators are derived.

Correlative amplitude–operational phase entanglement embodied by the EPR-pair eigenstate |η⟩

After a penetrating analysis of the known number–phase commutative relations we find that the common eigenvector |η of two particles' relative coordinate and total momentum also embodies the entanglement with respect to the correlative amplitude–operational phase. Based on the fact that the number-difference operator is the canonical conjugate to the Noh–Fougeres–Mandel operational phase operator, the corresponding entanglement is also briefly discussed.

Two-Mode Wigner Operator in 〈ξ| Representation and 〈τ| Representation

We derive the two-mode Wigner operator in the 〈ξ| representation and 〈τ| representation, where |ξ〉 is the common eigenvector of the mass-weighted relative coordinate and the mass-combinatorial momentum, and |τ〉 is the common eigenvector of the mass-weighted relative coordinate and the mass-combinatorial momentum. As an application, we calculate the Wigner function of some two-mode state.