Common Eigenvector Research Articles

A Comparison of Some Methods for the Selection of a Common Eigenvector Model for the Covariance Matrices of Two Groups

Two new nonparametric common principal component model selection procedures based on bootstrap distributions of the vector correlations of all combinations of the eigenvectors from two groups are proposed. The performance of these methods is compared in a simulation study to the two parametric methods previously suggested by Flury in 1988, as well as modified versions of two nonparametric methods proposed by Klingenberg in 1996 and then by Klingenberg and McIntyre in 1998. The proposed bootstrap vector correlation distribution (BVD) method is shown to outperform all of the existing methods in most of the simulated situations considered.

Fault Detection for Time-Varying Processes

In this brief, a new manifold learning method is proposed. Then, a process monitoring approach is proposed for handling the multimode monitoring problem in the electro-fused magnesia furnace based on the proposed manifold learning method. In the conventional methods, only partial common information is shared by different modes, i.e., the common eigenvectors. Compared with the conventional methods, the contributions are a new method of extracting the common subspace of different modes is proposed based on the manifold learning. The common subspace extracted by the proposed manifold learning method is shared by all different modes, and after those two different subspaces are separated, the common and specific subspace models are built and analyzed, respectively. The monitoring is carried out in the manifold subspaces.

Spatial eigenvector filtering for spatiotemporal crime mapping and spatial crime analysis

Spatial and spatiotemporal analyses are exceedingly relevant to determine criminogenic factors. The estimation of Poisson and negative binomial models (NBM) is complicated by spatial autocorrelation. Therefore, first, eigenvector spatial filtering (ESF) is introduced as a method for spatiotemporal mapping to uncover time-invariant crime patterns. Second, it is demonstrated how ESF is effectively used in criminology to invalidate model misspecification, i.e., residual spatial autocorrelation, using a nonviolent crime dataset for the metropolitan area of Houston, Texas, over the period 2005–2010. The results suggest that local and regional geography significantly contributes to the explanation of crime patterns. Furthermore, common space-time eigenvectors selected on an annual basis indicate striking spatiotemporal patterns persisting over time. The findings about the driving forces behind Houston’s crime show that linear and nonlinear, spatially filtered, NBMs successfully absorb latent autocorrelation and, therefore, prevent parameter estimation bias. The consideration of a spatial filter also increases the explanatory power of the regressions. It is concluded that ESF can be highly recommended for the integration in spatial and spatiotemporal modeling toolboxes of law enforcement agencies.

Generalized Shemesh criterion, common invariant subspaces and irreducible completely positive superoperators

Assume that are complex matrices. We give a computable criterion for existence of a common eigenvector of which generalize the result of D. Shemesh established for two matrices. We use this criterion to prove some necessary and sufficient condition for to have a common invariant subspace of dimension , , if every has pairwise different eigenvalues. Since the set of all matrices having multiple eigevalues has Lebesgue measure , the condition is sufficient in practical applications. Being motivated by quantum information theory, we give a flavour of such applications for irreducible completely positive superoperators.

Quantized consensus over directed networks with switching topologies

This paper studies the quantized consensus problem for a group of agents over directed networks with switching topologies. We propose an effective distributed protocol with an adaptive finite-level uniform quantized strategy, under which consensus among agents is guaranteed with weaker communication conditions. In particular, we analytically prove that each agent sending 5-level quantized information to each of its neighbors, together with 3-level quantized information to itself at each time step, which suffices for attaining consensus with an exponential convergence rate as long as the duration of all link failures in the directed network is bounded. By dropping the typical common left eigenvector requirement for the existence of common quadratic Lyapunov function, we conduct the convergence analysis based on the notion of input-to-output stability. The proposed quantized protocol has favorable merits of requiring little communication overhead and increasing robustness to link unreliability, and it fits well into the digital network framework.

Complex conductivity tensor of anisotropic hydrocarbon-bearing shales and mudrocks

A model was recently introduced to describe the complex electrical conductivity and high-frequency dielectric constant of isotropic clayey porous materials. We generalized that approach to the case of anisotropic and tight hydrocarbon-bearing shales and mudrocks by introducing tensorial versions of formation factor and tortuosity. In-phase and quadrature conductivity tensors have common eigenvectors, but the eigenvectors of the dielectric tensor may be different due to influence of the solid phase at high frequencies. In-phase and quadrature contributions to complex electrical conductivity depend on saturation, salinity, porosity, temperature, and cation exchange capacity (alternatively, specific surface area) of the porous material. Kerogen is likely to have a negligible contribution to the cation exchange capacity of the material because all exchangeable sites in the functional groups of organic matter may have been polymerized during diagenesis. An anisotropic experiment is performed to validate some of the properties described by the proposed model, especially to verify that the electrical anisotropy factor is the same for in-phase and quadrature conductivities. We used two samples from the Bakken formation. Experimental data confirm the validity of the model. Also, the range of values for cation exchange capacity determined when implementing the new model with experimental data agree with the known range of cation exchange capacity for the Bakken shale. Measurements indicate that the bulk-space tortuosity in the direction normal to bedding plane can be higher than 100.

On common eigenvectors for semigroups of matrices in tropical and traditional linear algebra

We prove the existence of a common eigenvector for commutative, nilpotent and quasinilpotent semigroups of matrices with complex or real nonnegative entries both in the conventional and tropical linear algebra.

On the existence of a common eigenvector for all matrices in the commutant of a single matrix

The main purpose of this paper is to study common invariant subspaces of any matrix in the centralizer of a given matrix A∈Mn(F), where F denotes an algebraically closed field. In particular, we obtain a necessary and sufficient condition for the existence of a common eigenvector for all the matrices in this set.

Pairwise commuting derivations of polynomial rings

We prove that n pairwise commuting derivations of the polynomial ring (or the power series ring) in n variables over a field k of characteristic 0 form a commutative basis of derivations if and only if they are k-linearly independent and have no common Darboux polynomials. This result generalizes a recent result due to Petravchuk and is an analogue of a well-known fact that a set of pairwise commuting linear operators on a finite dimensional vector space over an algebraically closed field has a common eigenvector.

Feedback Stabilization of Switching Discrete-Time Systems via Lie-Algebraic Techniques

This technical note addresses the stabilization of switching discrete-time linear systems with control inputs under arbitrary switching. A sufficient condition for the uniform global exponential stability (UGES) of such systems is the existence of a common quadratic Lyapunov function (CQLF) for the component subsystems, which is ensured when the closed-loop component subsystem matrices are stable and generate a solvable Lie algebra. The present work develops an iterative algorithm that seeks the feedback maps required for stabilization based on the previous Lie-algebraic condition. The main theoretical contribution of the technical note is to show that this algorithm will find the required feedback maps if and only if the Lie-algebraic problem has a solution. The core of the proposed algorithm is a common eigenvector assignment procedure, which is executed at every iteration. We also show how the latter procedure can be numerically implemented and provide a key structural condition which, if satisfied, greatly simplifies the required computations.

Diagonalization of matrices over the domain of principal ideals with minimal polynomial m(λ) = (λ − α)(λ − β), α ≠ β

The necessary and sufficient conditions of diagonalization of matrices over a domain of principal ideals with minimal polynomial m(λ) = (λ − α)(λ − β), α ≠ β are obtained. On the basis of the obtained results, the conditions under which the matrices have common eigenvectors are indicated.

SIFAT-SIFAT SPEKTRAL DAN STRUKTUR KOMBINATORIK PADA SISTEM POSITIF 2D

The dynamics of a 2D positive system depends on the pair of nonnegative square matrices thatprovide the updating of its local states. In this paper, several spectral properties, like finitememory, separablility and property L, which depend on the characteristic polynomial of thepair, are investigated under the nonnegativity constraint and in connection with thecombinatorial structure of the matrices.Some aspects of the Perron-Frobenius theory are extended to the 2D case; in particular,conditions are provided guaranteeing the existence of a common maximal eigenvector for twononnegative matrices with irreducible sum. Finally, some results on 2D positive realizationsare presented

Nonsmooth control design for stabilizing switched linear systems by left eigenstructure assignment

AbstractNovel design algorithms for exponential stability of switched linear systems using the left eigenstructure assignment approach by state feedback control are proposed in this article. For given switching constraints in the state space Rn, a state feedback controller for single-input switched systems is designed, based on the fact that closed loop system solutions are enforced to converge towards the invariant hyperplane attractor in Rn, as defined by the imposed common left eigenvector. The latter is appropriately constructed to guarantee the simultaneous stabilization of all constituent linear sub-systems, and to avoid the intersection of the switching manifolds with its invariant attractor set. For arbitrary switching, a set of n linearly independent common left eigenvectors in Rn is selected, introducing n hyperplane attractors, and n(n – 1) switching control hyperplane manifolds. The attrators are sequentially “turned on”, in accordance with the switching event generated by the Filippov solutions upon hitting the underlying control manifolds. Thereby, an additional state feedback control action for a second input is designed, using the nonsmooth Lyapunov stability criteria to avoid sliding modes and guarantee exponential stability for the Filippov solutions.

On commuting matrices in max algebra and in classical nonnegative algebra

This paper studies commuting matrices in max algebra and nonnegative linear algebra. Our starting point is the existence of a common eigenvector which directly leads to max analogs and nonnegative analogs of some classical results for complex matrices. We also investigate Frobenius normal forms of commuting matrices, particularly when the Perron roots of the components are distinct. For the case of max algebra, we show how the intersection of eigencones of commuting matrices can be described and we consider connections with Boolean algebra which enables us to prove that two commuting irreducible matrices in max algebra have a common eigennode.

The Cowin–Mehrabadi theorem for an axis of symmetry

The well-known Cowin–Mehrabadi Theorem deals with necessary and sufficient conditions for a normal n to a symmetry plane. Necessary conditions require that n be a common eigenvector of C ijkk , C ikjk and C ijkl n j n l . It is shown that a vector parallel to an axis of symmetry must also satisfy these conditions. An axis of rotational symmetry is also a normal to a plane of symmetry except in the case of a trigonal material. Being a common eigenvector of C ijkk and C ikjk belonging to a nondegenerate eigenvalue guarantees it to be an axis of symmetry.

On pairs of commuting derivations of the polynomial ring in one or two variables

It is well known that each pair of commuting linear operators on a finite dimensional vector space over an algebraically closed field has a common eigenvector. We prove an analogous statement for derivations of k [ x ] and k [ x , y ] over any field k of zero characteristic. In particular, if D 1 and D 2 are commuting derivations of k [ x , y ] and they are linearly independent over k, then either (i) they have a common polynomial eigenfunction; i.e., a nonconstant polynomial f ∈ k [ x , y ] such that D 1 ( f ) = λ f and D 2 ( f ) = μ f for some λ , μ ∈ k [ x , y ] , or (ii) they are Jacobian derivations D u ( g ) : = ∂ u ∂ x ∂ u ∂ y ∂ g ∂ x ∂ g ∂ y , D v ( g ) : = ∂ v ∂ x ∂ v ∂ y ∂ g ∂ x ∂ g ∂ y for all g ∈ k [ x , y ] defined by some u , v ∈ k [ x , y ] for which D u ( v ) is a nonzero constant.

Effective Way of Finding Common Eigenvectors of Center-of-Mass Coordinate and Mass-Weighted Relative Momenta of Multi-Partite via IWOP Technique

We present a convenient approach to finding multi-partite entangled state with continuum variables, which is the common eigenvectors of center-of-mass coordinate and mass-weighted relative momenta, by decomposing the normally ordered Gaussian-form operator expressing the completeness relation which is constructed by analyzing the eigenvector equations. The whole derivation is based on the technique of integration within an ordered product of operators.

New two-mode intermediate momentum-coordinate representation with quantum entanglement and its application

We construct four linear composite operators for a two-particle system and give common eigenvectors of those operators. The technique of integration within an ordered product (IWOP) of operators is employed to prove that those common eigenvectors are complete and orthonormal. Therefore, a new two-mode intermediate momentum-coordinate representation which involves quantum entanglement for a two-particle system is proposed and applied to some two-body dynamic problems. Moreover, the pure-state density matrix |ξ1, ξ2〉C,D C,D〈ξ1, ξ2| is a Radon transform of Wigner operator.

Eigenvectors of a rotation matrix

If a tensor is invariant under rotation about a fixed axis, the matrices representing the tensor and the rotation commute with each other. The two matrices have common eigenvectors, therefore a knowledge of eigenvectors of the rotation matrix provides us with some information about eigenvectors of the tensor. This result is applied to derive familiar representations of a transversely isotropic tensor of rank 2 and the elasticity tensor possessing tetragonal symmetry. Representation of the elasticity tensor belonging to a particular symmetry class can be achieved in an elegant manner.

MULTI-MODE LINEAR COMPOSITE MOMENTUM REPRESENTATION AND ITS APPLICATION

A unitary transformation matrix, n linear-composite coordinate operators and n linear-composite momentum operators are constructed for an n-particle system, and the complete and orthonormal common eigenvectors of the multi-mode linear composite momentum operators are examined by virtue of the technique of integration within an ordered product of operators. The multi-mode linear composite momentum representation is proposed, and its application to a general two-mode forced quantum harmonic oscillator system with kinetic coupling is presented for solving some dynamic problems.