Common Eigenvector Research Articles

Parameterized multi-observable sum uncertainty relations

The uncertainty principle is one of the fundamental features of quantum mechanics and plays an essential role in quantum information theory. We study uncertainty relations based on variance for arbitrary finite N quantum observables. We establish a series of parameterized uncertainty relations in terms of the parameterized norm inequalities, which improve the exiting variance-based uncertainty relations. The lower bounds of our uncertainty inequalities are non-zero unless the measured state is a common eigenvector of all the observables. Detailed examples are provided to illustrate the tightness of our uncertainty relations.

Common canonical variate analysis (CCVA) based modeling and monitoring for multimode processes

Multiple operating modes are common in industrial processes due to feed stock alterations, product specifications, working environment changes and so on. Although different modes show different behaviors, some underlying process characteristics may stay invariable as mode changes, which reveal the essence information of the process. In this work, the issue of multimode process monitoring is studied with subspace separation, in which each mode is divided into the common subspace and specific subspace. A modified canonical variate analysis (CVA), termed as common CVA (CCVA), is put forward to extract the mode-common features based on joint approximate diagonalization method. The concatenation of all Hankel matrices is analyzed to find a common orthonormal set eigenvectors by minimization of joint diagonality criterion. Then, the remaining part of each mode is regarded as local specific subspace, which provides more representative information in each different mode. CVA algorithm is applied to build multiple local models based on mode-specific information. Two case studies, a numerical example and Tennessee Eastman (TE) process, are provided to validate the feasibility and effectiveness of the proposed method in monitoring abnormal operation for multimode processes.

Completeness of Bethe Ansatz for Gaudin Models with 𝔤𝔩(1|1) Symmetry and Diagonal Twists

We studied the Gaudin models with gl(1|1) symmetry that are twisted by a diagonal matrix and defined on tensor products of polynomial evaluation gl(1|1)[t]-modules. Namely, we gave an explicit description of the algebra of Hamiltonians (Gaudin Hamiltonians) acting on tensor products of polynomial evaluation gl(1|1)[t]-modules and showed that a bijection exists between common eigenvectors (up to proportionality) of the algebra of Hamiltonians and monic divisors of an explicit polynomial written in terms of the highest weights and evaluation parameters. In particular, our result implies that each common eigenspace of the algebra of Hamiltonians has dimension one. We also gave dimensions of the generalized eigenspaces.

Finite orbits of monodromies of rank two Fuchsian systems

We classified finite orbits of monodromies of the Fuchsian system for five $2\times 2$ matrices. The explicit proof of this result is given. We have proposed a conjecture for a similar classification for $6$ or more $2\times 2$ matrices. Cases in which all monodromy matrices have a common eigenvector are excluded from the consideration. To classify the finite monodromies of the Fuchsian system we combined two methods developed in this paper. The first is an induction method: using finite orbits of smaller number of monodromy matrices the method allows the construction of such orbits for bigger numbers of matrices. The second method is a formalism for representing the tuple of monodromy matrices in a way that is invariant under common conjugation way, this transforms the problem into a form that allows one to work with rational numbers only. The classification developed in this paper can be considered as the first step to a classification of algebraic solutions of the Garnier system.

Construction of the spectral function from noncommuting spectral moment matrices

The LDA+U method is widely used to study the properties of realistic solids with strong electron correlations. One of its main shortcomings is that it does not provide direct access to the temperature dependence of material properties such as the Dzyaloshinskii-Moriya interaction, the anomalous Hall conductivity, and the spin-orbit torque. While the method of spectral moments allows us in principle to compute these quantities directly at finite temperatures, the standard two-pole approximation can be applied only to Hamiltonians that are effectively of single-band type. We do a first step to explore if the method of spectral moments may replace the LDA+U method in first-principles calculations of correlated solids with many bands in cases where the direct assessment of the temperature dependence of equilibrium and response functions is desired: The spectral moments of many-band Hamiltonians of correlated electrons do not commute and therefore they do not possess a system of common eigenvectors. We show that nevertheless the spectral function may be constructed from the spectral moments by solving a system of coupled non-linear equations. Additionally, we show how to compute the anomalous Hall conductivity of correlated electrons from this spectral function. We demonstrate the method for the Hubbard-Rashba model, where the standard two-pole approximation cannot be applied, because spin-orbit interaction (SOI)couples the spin-up and the spin-down bands. In the quest for new quantum states that arise from the combination of SOI and correlation effects, the Hartree-Fock approximation is frequently used to obtain a first approximation for the phase diagram. We propose that using the many-band generalization of the selfconsistent moment method instead of Hartree-Fock in such exploratory model calculations may improve the accuracy significantly, while keeping the computational burden low.

A novel procedure for constructing invariant subspaces of a set of matrices

A problem that is frequently encountered in a variety of mathematical contexts is to find the common invariant subspaces of a single or of a set of matrices. A new method is proposed that gives a definitive answer to this problem. The key idea consists of finding common eigenvectors for exterior powers of the matrices concerned. A convenient formulation of the Plücker relations is then used to ensure that these eigenvectors actually correspond to subspaces or provide the initial constraints for eigenvectors involving parameters. A procedure for computing the divisors of a totally decomposable vector is also provided. Several examples are given for which the calculations are too tedious to do by hand and are performed by coding the conditions found into Maple. Our main motivation lies in Lie symmetry, where the invariant subspaces of the adjoint representations for the Lie symmetry algebra of a differential equation must be known explicitly and comprehensively in order to determine all the ideals of the Lie symmetry algebra.

Loops of Infinite Order and Toric Foliations

In 2005 Dullin et al. proved that the non-zero vector of Maslov indices is an eigenvector with eigenvalue 1 of the monodromy matrices of an integrable Hamiltonian system. We take a close look at the geometry behind this result and extend it to a more general context. We construct a bundle morphism defined on the lattice bundle of an (general) integrable system, which can be seen as a generalization of the vector of Maslov indices. The non-triviality of this bundle morphism implies the existence of common eigenvectors with eigenvalue 1 of the monodromy matrices, and gives rise to a corank 1 toric foliation refining the original one induced by the integrable system. Furthermore, we show that in the case where the system has 2 degrees of freedom, this implies the global existence of a free S^{1} action.

Completeness of Bethe ansatz for Gaudin models associated with [formula omitted

We study the Gaudin models associated with gl(1|1). We give an explicit description of the algebra of Hamiltonians (Gaudin Hamiltonians) acting on tensor products of polynomial evaluation gl(1|1)[t]-modules. It follows that there exists a bijection between common eigenvectors (up to proportionality) of the algebra of Hamiltonians and monic divisors of an explicit polynomial written in terms of the highest weights and evaluation parameters. In particular, our result implies that each common eigenspace of the algebra of Hamiltonians has dimension one. Therefore, we confirm [7, Conjecture 8.3]. We also give dimensions of the generalized eigenspaces. Moreover, we express the generating pseudo-differential operator of Gaudin transfer matrices associated to antisymmetrizers in terms of the quadratic Gaudin transfer matrix and the center of U(gl(1|1)[t]).

DCT and DST Filtering With Sparse Graph Operators

Graph filtering is a fundamental tool in graph signal processing. Polynomial graph filters (PGFs), defined as polynomials of a fundamental graph operator, can be implemented in the vertex domain, and usually have a lower complexity than frequency domain filter implementations. In this paper, we focus on the design of filters for graphs with graph Fourier transform (GFT) corresponding to a discrete trigonometric transform (DTT), i.e., one of 8 types of discrete cosine transforms (DCT) and 8 discrete sine transforms (DST). In this case, we show that multiple sparse graph operators can be identified, which allows us to propose a generalization of PGF design: multivariate polynomial graph filter (MPGF). First, for the widely used DCT-II (type-2 DCT), we characterize a set of sparse graph operators that share the DCT-II matrix as their common eigenvector matrix. This set contains the well-known connected line graph. These sparse operators can be viewed as graph filters operating in the DCT domain, which allows us to approximate any DCT graph filter by a MPGF, leading to a design with more degrees of freedom than the conventional PGF approach. Then, we extend those results to all of the 16 DTTs as well as their 2D versions, and show how their associated sets of multiple graph operators can be determined. We demonstrate experimentally that ideal low-pass and exponential DCT/DST filters can be approximated with higher accuracy with similar runtime complexity. Finally, we apply our method to transform-type selection in a video codec, AV1, where we demonstrate significant encoding time savings, with a negligible compression loss.

Stepwise Covariance-Free Common Principal Components (CF-CPC) With an Application to Neuroscience.

Finding the common principal component (CPC) for ultra-high dimensional data is a multivariate technique used to discover the latent structure of covariance matrices of shared variables measured in two or more k conditions. Common eigenvectors are assumed for the covariance matrix of all conditions, only the eigenvalues being specific to each condition. Stepwise CPC computes a limited number of these CPCs, as the name indicates, sequentially and is, therefore, less time-consuming. This method becomes unfeasible when the number of variables p is ultra-high since storing k covariance matrices requires O(kp2) memory. Many dimensionality reduction algorithms have been improved to avoid explicit covariance calculation and storage (covariance-free). Here we propose a covariance-free stepwise CPC, which only requires O(kn) memory, where n is the total number of examples. Thus for n < < p, the new algorithm shows apparent advantages. It computes components quickly, with low consumption of machine resources. We validate our method CFCPC with the classical Iris data. We then show that CFCPC allows extracting the shared anatomical structure of EEG and MEG source spectra across a frequency range of 0.01–40 Hz.

On Eigenvectors of Nilpotent Lie Algebras of Linear Operators

We give a condition ensuring that the operators in a nilpotent Lie algebra of linear operators on a finite dimensional vector space have a common eigenvector.

On the spectral problem of the quantum KdV hierarchy

The spectral problem for the quantum dispersionless Korteweg–de Vries (KdV) hierarchy, aka the quantum Hopf hierarchy, is solved by Dubrovin. In this article, following Dubrovin, we study Buryak–Rossi’s quantum KdV hierarchy. In particular, we prove a symmetry property and a non-degeneracy property for the quantum KdV Hamiltonians. On the basis of this we construct a complete set of common eigenvectors. The analysis underlying this spectral problem implies certain vanishing identities for combinations of characters of the symmetric group. We also comment on the geometry of the spectral curves of the quantum KdV hierarchy and we give a representation of the quantum dispersionless KdV Hamiltonians in terms of multiplication operators in the class algebra of the symmetric group.

On the Supersymmetric XXX Spin Chains Associated to $$\mathfrak {gl}_{1|1}$$

We study the \(\mathfrak {gl}_{1|1}\) supersymmetric XXX spin chains. We give an explicit description of the algebra of Hamiltonians acting on any cyclic tensor products of polynomial evaluation \(\mathfrak {gl}_{1|1}\) Yangian modules. It follows that there exists a bijection between common eigenvectors (up to proportionality) of the algebra of Hamiltonians and monic divisors of an explicit polynomial written in terms of the Drinfeld polynomials. In particular our result implies that each common eigenspace of the algebra of Hamiltonians has dimension one. We also give dimensions of the generalized eigenspaces. We show that when the tensor product is irreducible, then all eigenvectors can be constructed using Bethe ansatz. We express the transfer matrices associated to symmetrizers and anti-symmetrizers of vector representations in terms of the first transfer matrix and the center of the Yangian.

Multitype weakly subcritical branching processes in random environment

Abstract A multi-type branching process evolving in a random environment generated by a sequence of independent identically distributed random variables is considered. The asymptotics of the survival probability of the process for a long time is found under the assumption that the matrices of the mean values of direct descendants have a common left eigenvector and the increment X of the associated random walk generated by the logarithms of the Perron roots of these matrices satisfies conditions E X &lt; 0 and E XeX &gt; 0.

Preliminary Multiple-Test Estimation, With Applications to k-Sample Covariance Estimation

Multisample covariance estimation—that is, estimation of the covariance matrices associated with k distinct populations—is a classical problem in multivariate statistics. A common solution is to base estimation on the outcome of a test that these covariance matrices show some given pattern. Such a preliminary test may, for example, investigate whether or not the various covariance matrices are equal to each other (test of homogeneity), or whether or not they have common eigenvectors (test of common principal components), etc. Since it is usually unclear what the possible pattern might be, it is natural to consider a collection of such patterns, leading to a collection of preliminary tests, and to base estimation on the outcome of such a multiple testing rule. In the present work, we therefore study preliminary test estimation based on multiple tests. Since this is of interest also outside k-sample covariance estimation, we do so in a very general framework where it is only assumed that the sequence of models at hand is locally asymptotically normal. In this general setup, we define the proposed estimators and derive their asymptotic properties. We come back to k-sample covariance estimation to illustrate the asymptotic and finite-sample behaviors of our estimators. Finally, we treat a real data example that allows us to show their practical relevance in a supervised classification framework.

A Survey on Multi-View Clustering.

Clustering is a machine learning paradigm of dividing sample subjects into a number of groups such that subjects in the same groups are more similar to those in other groups. With advances in information acquisition technologies, samples can frequently be viewed from different angles or in different modalities, generating multi-view data. Multi-view clustering, that clusters subjects into subgroups using multi-view data, has attracted more and more attentions. Although MVC methods have been developed rapidly, there has not been enough survey to summarize and analyze the current progress. Therefore, we propose a novel taxonomy of the MVC approaches. Similar to other machine learning methods, we categorize them into generative and discriminative classes. In discriminative class, based on the way of view integration, we split it further into five groups: Common Eigenvector Matrix, Common Coefficient Matrix, Common Indicator Matrix, Direct Combination and Combination After Projection. Furthermore, we relate MVC to other topics: multi-view representation, ensemble clustering, multi-task clustering, multi-view supervised and semi-supervised learning. Several representative real-world applications are elaborated for practitioners. Some benchmark multi-view datasets are introduced and representative MVC algorithms from each group are empirically evaluated to analyze how they perform on benchmark datasets. To promote future development of MVC approaches, we point out several open problems that may require further investigation and thorough examination.

Kerdock Codes Determine Unitary 2-Designs

The non-linear binary Kerdock codes are known to be Gray images of certain extended cyclic codes of length $N = 2^m$ over $\mathbb{Z}_4$. We show that exponentiating these $\mathbb{Z}_4$-valued codewords by $\imath \triangleq \sqrt{-1}$ produces stabilizer states, that are quantum states obtained using only Clifford unitaries. These states are also the common eigenvectors of commuting Hermitian matrices forming maximal commutative subgroups (MCS) of the Pauli group. We use this quantum description to simplify the derivation of the classical weight distribution of Kerdock codes. Next, we organize the stabilizer states to form $N+1$ mutually unbiased bases and prove that automorphisms of the Kerdock code permute their corresponding MCS, thereby forming a subgroup of the Clifford group. When represented as symplectic matrices, this subgroup is isomorphic to the projective special linear group PSL($2,N$). We show that this automorphism group acts transitively on the Pauli matrices, which implies that the ensemble is Pauli mixing and hence forms a unitary $2$-design. The Kerdock design described here was originally discovered by Cleve et al. (arXiv:1501.04592), but the connection to classical codes is new which simplifies its description and translation to circuits significantly. Sampling from the design is straightforward, the translation to circuits uses only Clifford gates, and the process does not require ancillary qubits. Finally, we also develop algorithms for optimizing the synthesis of unitary $2$-designs on encoded qubits, i.e., to construct logical unitary $2$-designs. Software implementations are available at https://github.com/nrenga/symplectic-arxiv18a, which we use to provide empirical gate complexities for up to $16$ qubits.

Stability analysis of fractional-order linear neutral delay differential\u2013algebraic system described by the Caputo\u2013Fabrizio derivative

This paper is concerned with the asymptotic stability of linear fractional-order neutral delay differential–algebraic systems described by the Caputo–Fabrizio (CF) fractional derivative. A novel characteristic equation is derived using the Laplace transform. Based on an algebraic approach, stability criteria are established. The effect of the index on such criteria is analyzed to ensure the asymptotic stability of the system. It is shown that asymptotic stability is ensured for the index-1 problems provided that a stability criterion holds for any delay parameter. Also, asymptotic stability is still valid for higher-index problems under the conditions that the system matrices have common eigenvectors and each pair of such matrices is simultaneously triangularizable so that a stability criterion holds for any delay parameter. An example is provided to demonstrate the effectiveness and applicability of the theoretical results.

On Razamat’s A2 and A3 kernel identities

In recent work on superconformal quantum field theories, Razamat arrived at elliptic kernel identities of a new and striking character: they relate solely to the root systems A2 and A3 and have no coupling type parameters, Razamat S S (2019 Lett. Math. Phys. 1091377–95). The pertinent two- and three-variable Hamiltonians are analytic difference operators and the kernel functions are built from the elliptic gamma function. Razamat presented compelling evidence for the validity of these identities, and checked them to a certain order in a power series expansion. This paper is mainly concerned with analytical proofs of these identities. More specifically, we furnish a complete proof of the identities for the A2 elliptic case and for the A3 hyperbolic case, and consider several specializations. We also discuss the implications the kernel identities might have for a Hilbert space scenario involving common eigenvectors of the Hamiltonians and the integral operators associated with the kernel functions.

A Relation Between Moore-Penrose Inverses of Hermitian Matrices and Its Application in Electrical Networks

A novel relation between the Moore-Penrose inverses of two nullity-1 $n\times n$ Hermitian matrices which share a common null eigenvector is established, and its application in electrical networks is illustrated by applying the result to Laplacian matrices of graphs.