Off-diagonal Entries Research Articles

Inseparable Gershgorin discs and the existence of conjugate complex eigenvalues of real matrices

We investigate the converse of the known fact that if the Gershgorin discs of a real n-by-n matrix may be separated by positive diagonal similarity, then the eigenvalues are real. In the 2-by-2 case, with appropriate signs for the off-diagonal entries, we find that the converse is correct, which raises several questions. First, in the 3-by-3 case, the converse is not generally correct, but, empirically, it is frequently true. Then, in the n-by-n case, n ≥ 3 , we find that if all the 2-by-2 principal submatrices have inseparable discs (‘strongly inseparable discs’), the full matrix must have a nontrivial pair of conjugate complex eigenvalues (i.e. cannot have all real eigenvalues). This hypothesis cannot generally be weakened.

A test for the identity of a high-dimensional correlation matrix based on the [formula omitted]-norm

This paper presents a new test procedure for the identity of a correlation matrix in high-dimensional asymptotic frameworks, the statistic of which is based on the ℓ4-norm of off-diagonal entries of the covariance matrix. A striking finding is that this test can achieve full power against both dense alternatives and sparse ones. Simulation results demonstrate its superiority over some competitors in various cases.

Reduced transfer operators for singular difference equations

For tridiagonal block Jacobi operators, the standard transfer operator techniques only work if the off-diagonal entries are invertible. Under suitable assumptions on the range and kernel of these off-diagonal operators which assure a homogeneous minimal coupling between the blocks, it is shown how to construct reduced transfer operators that have the usual Krein space unitarity property and also a crucial monotonicity in the energy variable. This allows to extend the results of oscillation theory to such systems.

Inverse eigenvalue and related problems for hollow matrices described by graphs

A hollow matrix described by a graph $G$ is a real symmetric matrix having all diagonal entries equal to zero and with the off-diagonal entries governed by the adjacencies in $G$. For a given graph $G$, the determination of all possible spectra of matrices associated with $G$ is the hollow inverse eigenvalue problem for $G$. Solutions to the hollow inverse eigenvalue problems for paths and complete bipartite graphs are presented. Results for related subproblems such as possible ordered multiplicity lists, maximum multiplicity of an eigenvalue, and minimum number of distinct eigenvalues are presented for additional families of graphs.

Assessment of Covariance Selection Methods in High-Dimensional Gaussian Graphical Models

The covariance selection in Gaussian graphical models consists in selecting, based on a sample of a multivariate normal vector, all those pairs of variables that are conditionally dependent given the remaining variables. This problem is equivalent to estimate the graph identifying the nonzero elements on the off-diagonal entries of the precision matrix. There are different proposals to carry out covariance selection in high-dimensional Gaussian graphical models, such as neighborhood selection and Glasso, among others. In this paper we introduce a methodology for evaluating the performance of graph estimators, defining the notion of non-informative estimator. Through a simulation study, the empirical behavior of Glasso in different structures of the precision matrix is investigated and its performance is analyzed according to different degrees of density of the graph. Our proposal can be used for other covariance selection methods.

Functions of the Laplacian Matrix With Application to Distributed Formation Control

In this article, we study a class of matrix functions of the combinatorial Laplacian that preserve its structure, i.e., that define matrices, which are positive semidefinite and which have zero row-sum and nonpositive off-diagonal entries. This formulation has the merit of presenting different incarnations of the Laplacian matrix that have appeared in recent literature in a unified framework. For the first time, we apply this family of Laplacian functions to consensus theory, and we show that they leave the agreement value unchanged and offer distinctive advantages in terms of performance and design flexibility. The theory is illustrated via worked examples and numerical experiments featuring four representative Laplacian functions in a shape-based distributed formation control strategy for single-integrator robots.

A central limit theorem for star-generators of S∞, which relates to traceless CCR-GUE matrices

We prove a limit theorem concerning the sequence of star-generators of [Formula: see text], where the expectation functional is provided by a character of [Formula: see text] with weights [Formula: see text] in the Thoma classification. The limit law turns out to be the law of a “traceless CCR-GUE” matrix, an analogue of the traceless GUE where the off-diagonal entries [Formula: see text] satisfy the commutation relation [Formula: see text]. The special case [Formula: see text] yields the law of a bona fide traceless GUE matrix, and we retrieve a result of Köstler and Nica from 2021, which in turn extended a result of Biane from 1995.

A new perspective on cooperative control of multi-agent systems through different types of graph Laplacians

In the field of systems and control, many cooperative control problems of multi-agent systems have been actively studied in the past two decades. This article aims to organize extensive existing work on different cooperative control problems into three categories, based on three different types of graph Laplacian matrices involved. A Laplacian matrix is an important representation of graph topology, and depending on the field of its entries, there are three types: ordinary Laplacian (non-negative diagonal entries and non-positive off-diagonal entries), signed Laplacian (arbitrary real entries), and complex Laplacian (arbitrary complex entries). Each type of graph Laplacian is useful in modeling and solving a different set of cooperative control problems. In particular, their algebraic properties are fundamental in characterizing stability and performance of the respective solution algorithms. To our best knowledge, organizing the literature on multi-agent cooperative control through the lens of different graph Laplacians is new.

Quasi-balanced weighing matrices, signed strongly regular graphs and association schemes

A weighing matrix W is quasi-balanced if |W||W|⊤=|W|⊤|W| has at most two off-diagonal entries, where |W|ij=|Wij|. A quasi-balanced weighing matrix W signs a strongly regular graph if |W| coincides with its adjacency matrix. Among other things, signed strongly regular graphs and their equivalent association schemes are presented.

An analysis of the perturbation error bounds for singular and eigenvalues of diagonally dominant matrices

In this paper, we investigate the error bounds of singular values and eigenvalues for diagonally dominant Hermite matrices when the matrix on its off-diagonal entries and on diagonally dominant part is perturbed with bound ε. Our results show that not only is the bound independent of the condition number of the matrix, but the matrix itself need not be symmetric positive definite. Furthermore, the bounds are applicable to both the eigenvalue and singular value of the matrix and can be used in generalized eigenvalue problems.

Rotational and translational diffusion of liquid n-hexane: EFP-based molecular dynamics analysis.

Molecular Dynamics (MD) simulations based on the Effective Fragment Potential (EFP) method are utilized to provide a comprehensive assessment of diffusion in liquid n-hexane. We decompose translational diffusion into components along and orthogonal to the long axis of the molecule. Rotational diffusion is decomposed into tumbling and spinning motions about this axis. Our analysis yields four corresponding diffusion coefficients which are related to diagonal entries in the complete 6 × 6 diffusion tensor accounting for the three rotational and three translational degrees of freedom and for the potential coupling between them. However, coupling between different degrees of freedom is expected to be minimal for a natural choice of the molecular body-fixed axis, so then off-diagonal entries in the tensor are negligible. This expectation is supported by a hydrodynamic analysis of the diffusion tensor which treats the liquid surrounding the molecule being tracked as a viscous continuum. Thus, the EFP MD analysis provides a comprehensive characterization of diffusion and also reveals expected shortcomings of the hydrodynamic treatment, particularly for rotational diffusion, when applied to neat liquids.

The homology of SL2 of discrete valuation rings

Let A be a discrete valuation ring with field of fractions F and residue field k such that |k|≠2,3,4,5,7,8,9,16,27,32,64. We prove that there is a natural exact sequenceH3(SL2(A),Z[12])→H3(SL2(F),Z[12])→RP1(k)[12]→0, where RP1(k) is the refined scissors congruence group of k. Let Γ0(mA) denote the congruence subgroup consisting of matrices in SL2(A) whose lower off-diagonal entry lies in the maximal ideal mA. We also prove that there is an exact sequence0→P‾(k)[12]→H2(Γ0(mA),Z[12])→H2(SL2(A),Z[12])→I2(k)[12]→0, where I2(k) is the second power of the fundamental ideal of the Grothendieck-Witt ring GW(k) and P‾(k) is a certain quotient of the scissors congruence group (in the sense of Dupont-Sah) P(k) of k.

On the minimum number of distinct eigenvalues of a threshold graph

For a graph G, we associate a family of real symmetric matrices, S(G), where for any A∈S(G), the location of the nonzero off-diagonal entries of A are governed by the adjacency structure of G. Let q(G) be the minimum number of distinct eigenvalues over all matrices in S(G). In this work, we give a characterization of all connected threshold graphs G with q(G)=2. Moreover, we study the values of q(G) for connected threshold graphs with trace 2, 3, n−2, n−3, where n is the order of threshold graph. The values of q(G) are determined for all connected threshold graphs with 7 and 8 vertices with two exceptions. Finally, a sharp upper bound for q(G) over all connected threshold graph G is given.

Real symmetric matrices and their negative eigenvalues

For two integers k≧0 and q≧1, consider symmetric matrices M with k negative eigenvalues counted with multiplicities and q pairwise distinct values of entries such that the rows of M are mutually distinct and the largest diagonal entry of M is less than or equal to the smallest off-diagonal entry of M. It is shown that the number of such matrices is finite when k and q are fixed. This generalizes some known results on the adjacency matrices of graphs. It is conjectured that any twin-free graph on n vertices with no isolated vertices has at least −1+log2⁡(n+2) negative adjacency eigenvalues.

Computing Different Realizations of Linear Dynamical Systems with Embedding Eigenvalue Assignment

In this paper we investigate realizability of discrete time linear dynamical systems (LDSs) in fixed state space dimension. We examine whether there exist different Θ = (A,B,C,D) state space realizations of a given Markov parameter sequence Y with fixed B, C and D state space realization matrices. Full observation is assumed in terms of the invertibility of output mapping matrix C. We prove that the set of feasible state transition matrices associated to a Markov parameter sequence Y is convex, provided that the state space realization matrices B, C and D are known and fixed. Under the same conditions we also show that the set of feasible Metzler-type state transition matrices forms a convex subset. Regarding the set of Metzler-type state transition matrices we prove the existence of a structurally unique realization having maximal number of non-zero off-diagonal entries. Using an eigenvalue assignment procedure we propose linear programming based algorithms capable of computing different state space realizations. By using the convexity of the feasible set of Metzler-type state transition matrices and results from the theory of non-negative polynomial systems, we provide algorithms to determine structurally different realization. Computational examples are provided to illustrate structural non-uniqueness of network-based LDSs.

The p-norm of circulant matrices via Fourier analysis

Abstract A recent work derived expressions for the induced p-norm of a special class of circulant matrices A(n, a, b) ∈ ℝ n × n , with the diagonal entries equal to a ∈ ℝ and the off-diagonal entries equal to b ≥ 0. We provide shorter proofs for all the results therein using Fourier analysis. The key observation is that a circulant matrix is diagonalized by a DFT matrix. The results comprise an exact expression for ǁAǁ p , 1 ≤ p ≤ ∞, where A = A(n, a, b), a ≥ 0 and for ǁAǁ2 where A = A(n, −a, b), a ≥ 0; for the other p-norms of A(n, −a, b), 2 < p < ∞, upper and lower bounds are derived.

Hereditarity of potential matrices and positive affine prediction of nonnegative risks from mixture models

ABSTRACT Nonnegative linear filtering of nonnegative risk variables necessitates positivity properties on the variance–covariance matrices of random effects, if experience rating is derived from mixture models. A variance–covariance matrix is a potential if it is nonsingular and if its inverse is diagonally dominant, with off-diagonal entries that are all nonpositive. We consider risk models with stationary random effects whose variance–covariance matrices are potentials. Positive credibility predictors of nonnegative risks are obtained from these mixture models. The set of variance–covariance matrices that are potentials is closed under extraction of principal submatrices. This strong hereditary property maintains the positivity of the affine predictor if the horizon is greater than one and if the history is lacunary. The specifications of the dynamic random effects presented in this paper fulfill the required positivity properties, and encompass the three possible levels for the length of memory in the mixing distribution. A case study discusses the possible strategies in the prediction of the pure premium from dynamic random effects.

On the forcing matching numbers of prisms of graphs

Let G be a graph on n vertices. Denote by f(G) the minimum size of a matching M in G such that M is uniquely extendable to a perfect matching in G. The prism of G is defined as G□K2, that is, the Cartesian product of G with the complete graph on 2 vertices. Assume that there exists an involutory matrix A with zero diagonal such that the graph described by the zero/non-zero pattern of off-diagonal entries of A is isomorphic to G. It is shown that f(G□K2)=n2 if G is bipartite. This generalizes a known result. The quantity f for prisms of some families of graphs are determined.

An inverse formula for the distance matrix of a fan graph

Let be the fan graph with vertices. The distance between any two distinct vertices i and j of is the length of the shortest path connecting i and j. Let be the symmetric matrix with diagonal entries equal to zero and off-diagonal entries equal to . In this paper, we find two positive semidefinite matrices and such that , all row sums of and are equal to zero, and find two rank one matrices and such that The interlacing property between the eigenvalues of and (resp., ) is also established.

Direct characterization of coherence of quantum detectors by sequential measurements

The quantum properties of quantum measurements are indispensable resources in quantum information processing and have drawn extensive research interest. The conventional approach to reveal the quantum properties relies on the reconstruction of the entire measurement operators by quantum detector tomography. However, many specific properties can be determined by a part of matrix entries of the measurement operators, which provides us the possibility to simplify the process of property characterization. Here, we propose a general framework to directly obtain individual matrix entries of the measurement operators by sequentially measuring two non-compatible observables. This method allows us to circumvent the complete tomography of the quantum measurement and extract the useful information for our purpose. We experimentally implement this scheme to monitor the coherent evolution of a general quantum measurement by determining the off-diagonal matrix entries. The investigation of the measurement precision indicates the good feasibility of our protocol to the arbitrary quantum measurements. Our results pave the way for revealing the quantum properties of quantum measurements by selectively determining the matrix entries of the measurement operators.