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Let G be a connected graph of order n , A ( G ) the adjacency matrix of G and D ( G ) the diagonal matrix of the row-sums of A ( G ) . For every real α ∈ [ 0 , 1 ] , Nikiforov defined the matrix A α ( G ) as A α ( G ) = α D ( G ) + ( 1 − α ) A ( G ) . In this paper, we obtain a factorization of the A α -characteristic polynomial of the some families of graphs and determine in these families the conditions about α so that A α ( G ) is positive semidefinite.

This article is devoted to the investigation of reduced-order dissipative filtering for Takagi-Sugeno (T-S) fuzzy Markov jump systems with the event-triggered mechanism. For the proposed event-triggered mechanism, its threshold parameter is constructed as a special diagonal matrix which can improve system performance by flexibly adjusting the matrix elements. Due to the impact of the sampling behaviors and the environmental disturbance, the asynchronization between the filter and the estimated system is considered in this article, which can be characterized by the hidden Markov model. Through handling the linear matrix inequalities (LMIs) with some slack matrices, event-triggered fuzzy filters are designed to guarantee the resulting system is stochastically stable and strictly dissipative. The proposed filter parameters are obtained by solving LMIs. Ultimately, both the effectiveness and advantages of the proposed reduced-order filter with the event-triggered mechanism are verified by a practical example.

By employing off-diagonal matrix Mittag-Leffler functions and stability theory for line fractional functional differential equations, a new technique is proposed to investigate the existence, uniqueness and global asymptotical stability of S -asymptotically periodic solution for a class of semilinear Caputo fractional functional differential equations. Some better results are derived, which improve and extend the existing research findings in recent years. As an application of the general theory, some decision theorems are established for the asymptotically dynamical behaviors for fractional four-neuron BAM neural networks. The methods used in this paper could be applied to the study of other fractional differential systems in the areas of science and engineering.

For a graph G , the signless Laplacian matrix Q ( G ) is defined as Q ( G ) = D ( G ) + A ( G ) , where A ( G ) is the adjacency matrix of G and D ( G ) the diagonal matrix whose main entries are the degrees of the vertices in G . The Q -spectrum of G is that of Q ( G ) . In the present paper, we are interested in the minimum values of the second largest signless Laplacian eigenvalue q 2 ( G ) of a connected graph G . We find the five smallest values of q 2 ( G ) over the set of connected graphs G with given order n . We also characterize the corresponding extremal graphs.

In this paper, we introduce the class of diagonally dominant with respect to a given LMI region D ⊂ C matrices. They are shown to possess the analogues of well-known properties of (classical) diagonally dominant matrices, e.g. their spectra are localized inside the region D . Moreover, we show that in some cases, diagonal D -dominance implies ( D , D ) -stability (i.e. the preservation of matrix spectra localization under multiplication by a positive diagonal matrix).

Let G be a connected graph, we define D Q ( G ) = T r ( G ) + D ( G ) as distance signless Laplacian matrix of G, where T r ( G ) and D ( G ) are diagonal matrix with vertex transmissions of G and distance matrix of G, respectively. In this paper, we characterize the extremal graphs which maximize the D Q -spectral radius among complements of unicyclic graphs and trees, respectively. And we also characterize the unique graph among complements of unicyclic graphs of diameter three which maximize the least D Q -eigenvalues.

For 0 ≤ α ≤ 1 , the A α -matrix of graph G is defined as A α ( G ) = α D ( G ) + ( 1 − α ) A ( G ) , where D ( G ) and A ( G ) are the diagonal matrix of the degrees and the adjacency matrix of G, respectively. On the premise of a fixed number of edges, Zhai et al. (2020) [15] depicted the extremal graph which has the largest A 1 2 -spectral radius, and also characterized the graph maximizing the A 1 2 -spectral radius among all graphs with given clique number (respectively, chromatic number). In this paper, we extend the conclusion of them to A α -spectral radius for α ∈ [ 1 2 , 1 ] .

The topic of finite-time stability criterion for neural networks with time-varying delays via a new argument Lyapunov-Krasovskii functional (LKF) was proposed and the time-varying delay of the system is without differentiable. For sufficient conditions of this study, a new (LKF) is combined with improved triple integral terms, namely the functionality of finite-time stability, integral inequality, and a positive diagonal matrix without using a free weighting matrix. The improved finite-time sufficient conditions for the neural network with time varying delay are given in terms of linear matrix inequalities (LMIs) and the results show improvement on previous studies. Numerical examples are given to illustrate the effectiveness of the proposed method.

Let G be a simple connected graph with n vertices. The transmission Tv of a vertex v is defined to be the sum of the distances from v to all other vertices in G, that is, Tv = Σu∈Vduv, where duv denotes the distance between u and v. Let T1,…, Tn be the transmission sequence of G. Let $$\cal{D}=(d_{ij})_{n\times n}$$ be the distance matrix of G, and $$\cal{T}$$ be the transmission diagonal matrix diag(T1,…,Tn). The matrix $$\cal{Q}(G)=\cal{T}+\cal{D}$$ is called the distance signless Laplacian of G. In this paper, we provide the distance signless Laplacian spectrum of complete k-partite graph, and give some sharp lower and upper bounds on the distance signless Laplacian spectral radius $$\mathbb{q}(G)$$ .

Distance metric learning (DML) aims to learn distance metrics that reflect the interactions between features and labels. Due to the high computational complexity, existing DML models are unsuitable for large-scale datasets. This study proposes a DML approach for large-scale problems by reducing the number of variables, utilizing sparse structures of the optimization problems, and taking advantage of large-scale computation platforms. The proposed approach treats DML as a linear space transformation problem and suggests that a full DML matrix can be approximated by a diagonal matrix in many cases. We solve the diagonal DML problem along with its l 1 and l 2 regularizations via linear and quadratic programming. To facilitate large-scale learning problems, we design a MapReduce framework to build triplets, which are encapsulations of triple data points used for the optimization problem, and develop a weighting mechanism for triplets according to their contributions to the whole distance distortion. Experiments show that the proposed approach is fast in large-scale DML applications with comparable accuracy to much more time-consuming full matrix models. Since the approach is implemented with the Scala language based on the Spark platform, it can be used directly by productive Java applications, which makes it highly practical for large-scale datasets.

Resistive crossbars implement parallel vector-matrix multiplication (VMM) in analog fashion, and thus enable fast and energy-efficient neuromorphic systems. However, interconnect resistance and resistive switching devices form a complex resistance network with sneak paths. It could result in severe distortions on the output currents. When implementing neural networks, current distortions also cause significant accuracy loss. This article proposes an accurate and computationally efficient model of VMM in resistive crossbars, called diagonal matrix regression (DMR), and incorporates the model into the topology of neural networks as DMR layer (DMRL). Given an ${m}{\times }{n}$ crossbar, two diagonal matrices are calculated directly according to the resistance network in a time complexity of only ${O}({m}^{2}+{n}^{2})$ . No hyper-parameter needs to be determined manually. Modeling of VMM is implemented in a time complexity of only ${O}({mn})$ . DMRL is developed to replace the weight matrix of neural networks so that the effect of interconnect resistance and the sneak path problem are well handled during ex-situ training. Using this technique, for the task of MNIST and fashion-MNIST classification, the accuracy is dramatically restored.

In the block version of the classical two-sided Jacobi method for the Hermitian eigenvalue problem, the off-diagonal elements of iterated matrix A ( k ) converge to zero. However, this fact alone does not necessarily guarantee that A ( k ) converges to a fixed diagonal matrix. The same is true for the matrix of accumulated unitary transformations Q ( k ) . We prove that under certain assumptions A ( k ) indeed converges to a fixed diagonal matrix, whose diagonal elements are the eigenvalues of the input matrix A. Next it is shown that for a simple eigenvalue the corresponding column of Q ( k ) converges to the corresponding eigenvector. For a multiple eigenvalue or a cluster of eigenvalues, we prove that the orthogonal projectors constructed from the corresponding columns of Q ( k ) converge to the orthogonal projector onto the eigenspace corresponding to those eigenvalues. Moreover, the appropriate convergence bounds are obtained for all discussed cases. Convergence results are also valid for the parallel block-Jacobi method with dynamic ordering. The developed theory is illustrated by numerical example.

Most existing super-resolution imaging methods fail to work in low signal-to-noise ratio (SNR) condition due to the ill-posed antenna measurement matrix, but the sparse-truncated singular value decomposition (TSVD) method can effectively suppress noise and improve azimuth resolution in low SNR condition. However, the current sparse-TSVD method encounters large computation cost, resulting in a slow algorithm speed. In this work, a fast sparse-TSVD super-resolution imaging method of real aperture radar is proposed. First, the proposed method is based on the results of TSVD, using the truncated unitary matrix and diagonal matrix to reconstruct the signal convolution model. The dimension of the reconstructed antenna measurement matrix reduces from $N \times N$ to $k \times N$ , and the dimension of the reconstructed echo matrix reduces from $N \times 1$ to $k \times 1$ , where $N$ is azimuth sampling points and $k$ is truncation parameter, $N \gg k$ . Much of the expensive matrix– multiplication computation can then be performed on the smaller matrices, thereby accelerating the algorithm. Second, an objective function is established as the ${l_{1}}$ constraint based on the regularization strategy. Lastly, this article employs iterative reweighted least square (IRLS) method to solve the objective function, and the dimension of the reversed matrix is lessened from $N \times N$ to $k \times k$ , speeding up the algorithm further. The simulation and real data verify that the proposed algorithm not only improves the azimuth resolution in low SNR condition but also increases computational efficiency compared with the sparse-TSVD method.

We address the structure identification and the uniform approximation of two fully nonlinear layer neural networks of the type$$f(x)=1^T h(B^T g(A^T x))$$f(x)=1Th(BTg(ATx))on$$\mathbb R^d$$Rd, where$$g=(g_1,\dots , g_{m_0})$$g=(g1,⋯,gm0),$$h=(h_1,\dots , h_{m_1})$$h=(h1,⋯,hm1),$$A=(a_1|\dots |a_{m_0}) \in \mathbb R^{d \times m_0}$$A=(a1|⋯|am0)∈Rd×m0and$$B=(b_1|\dots |b_{m_1}) \in \mathbb R^{m_0 \times m_1}$$B=(b1|⋯|bm1)∈Rm0×m1, from a small number of query samples. The solution of the case of two hidden layers presented in this paper is crucial as it can be further generalized to deeper neural networks. We approach the problem by sampling actively finite difference approximations to Hessians of the network. Gathering several approximate Hessians allows reliably to approximate the matrix subspace$$\mathcal W$$Wspanned by symmetric tensors$$a_1 \otimes a_1,\dots ,a_{m_0}\otimes a_{m_0}$$a1⊗a1,⋯,am0⊗am0formed by weights of the first layer together with the entangled symmetric tensors$$v_1 \otimes v_1 ,\dots ,v_{m_1}\otimes v_{m_1}$$v1⊗v1,⋯,vm1⊗vm1, formed by suitable combinations of the weights of the first and second layer as$$v_\ell =A G_0 b_\ell /\Vert A G_0 b_\ell \Vert _2$$vℓ=AG0bℓ/‖AG0bℓ‖2,$$\ell \in [m_1]$$ℓ∈[m1], for a diagonal matrix$$G_0$$G0depending on the activation functions of the first layer. The identification of the 1-rank symmetric tensors within$$\mathcal W$$Wis then performed by the solution of a robust nonlinear program, maximizing the spectral norm of the competitors constrained over the unit Frobenius sphere. We provide guarantees of stable recovery under a posteriori verifiable conditions. Once the 1-rank symmetric tensors$$\{a_i \otimes a_i, i\in [m_0]\}\cup \{v_\ell \otimes v_\ell , \ell \in [m_1] \}$${ai⊗ai,i∈[m0]}∪{vℓ⊗vℓ,ℓ∈[m1]}are computed, we address their correct attribution to the first or second layer ($$a_i$$ai’s are attributed to the first layer). The attribution to the layers is currently based on a semi-heuristic reasoning, but it shows clear potential of reliable execution. Having the correct attribution of the$$a_i,v_\ell $$ai,vℓto the respective layers and the consequent de-parametrization of the network, by using a suitably adapted gradient descent iteration, it is possible to estimate, up to intrinsic symmetries, the shifts of the activations functions of the first layer and compute exactly the matrix$$G_0$$G0. Eventually, from the vectors$$v_\ell =A G_0 b_\ell /\Vert A G_0 b_\ell \Vert _2$$vℓ=AG0bℓ/‖AG0bℓ‖2’s and$$a_i$$ai’s one can disentangle the weights$$b_\ell $$bℓ’s, by simple algebraic manipulations. Our method of identification of the weights of the network is fully constructive, with quantifiable sample complexity and therefore contributes to dwindle the black box nature of the network training phase. We corroborate our theoretical results by extensive numerical experiments, which confirm the effectiveness and feasibility of the proposed algorithmic pipeline.

Given any real α ∈ [ 0 , 1 ] , the α-index of a graph G is the largest eigenvalue λ α ( G ) of the matrix A α ( G ) = α D ( G ) + ( 1 − α ) A ( G ) , where A ( G ) and D ( G ) stand for the adjacency matrix and the diagonal matrix of vertex degrees in G, respectively. In this paper, we determine the unique graph with maximum α-index among all graphs of order n with no 4-cycle and 5-cycle, respectively. These results partially answer a proposed problem in Nikiforov (2017) [7, Problem 24] .

Let A(G) and D(G) be the adjacency matrix and the degree diagonal matrix of a graph G, respectively. For any real α∈[0,1], Nikiforov [Appl Anal Discrete Math. 2017;11:81–107] proposed the matrix Aα...

We consider a linear control system with an almost periodic coefficient matrix whose mean is a diagonal matrix and with a linear state feedback control. The control matrix is a constant square matrix and has zero rows, and its nonzero rows are linearly independent. The feedback gain is assumed to be almost periodic and such that its frequency module is contained in the frequency module of the coefficient matrix. Necessary and sufficient conditions under which the asynchronous spectrum assignment problem is solvable for this system are obtained. The problem is to find a feasible control such that the closed-loop system has almost periodic solutions with prescribed frequencies, the intersection of the frequency modules of the solution and the coefficient matrix being trivial.

This brief paper proposes a robust generalized proportionate diffusion Least Mean Square (LMS) algorithm for distributed estimation of a parameter vector in a network. The contribution of this brief is twofold. First, we generalize the concept of proportionate diffusion LMS by letting the gain matrix to be non-diagonal instead of being a diagonal matrix. Second, to achieve robustness to impulsive noise while simultaneously maintaining a fast convergence property, we use a combination of Mean Square Deviation (MSD) and disturbance incurred in the adaptation step as the objective cost function. By simplifying and optimizing the proposed cost function, a closed form formula is obtained for the gain matrix in the general non-diagonal case. Simulation results demonstrate the efficiency of the proposed method in comparison to some other state-of-the-art algorithms.

The covariance matrix, which should be estimated from the data, plays an important role in many multivariate procedures, and its positive definiteness (PDness) is essential for the validity of the procedures. Recently, many regularized estimators have been proposed and shown to be consistent in estimating the true matrix and its support under various structural assumptions on the true covariance matrix. However, they are often not PD. In this paper, we propose a simple modification to make a regularized covariance matrix be PD while preserving its support and the convergence rate. We focus on the matrix $$\ell_{\infty }$$ norm error in covariance matrix estimation because it could allow us to bound the error in the downstream multivariate procedure relying on it. Our proposal in this paper is an extension of the fixed support positive-definite (FSPD) modification by Choi et al. (2019) from spectral and Frobenius norms to the matrix $$\ell_{\infty }$$ norm. Like the original FSPD, we consider a convex combination between the initial estimator (the regularized covariance matrix without PDness) and a given form of the diagonal matrix minimize the $$\ell_{\infty }$$ distance between the initial estimator and the convex combination, and find a closed-form expression for the modification. We apply the procedure to the minimum variance portfolio (MVP) optimization problem and show that the vector $$\ell_{\infty }$$ error in the estimation of the optimal portfolio weight is bounded by the matrix $$\ell _{\infty }$$ error of the plug-in covariance matrix estimator. We illustrate the MVP results with S&P 500 daily returns data from January 1978 to December 2014.

We investigate geometric properties of algebraic sets defined by matrix pairs with special commutators. Specifically, we prove that the set consisting of the n × n matrix pairs whose commutators are diagonal matrices is a complete intersection with two components, one of which is the so called commuting variety that consists of commuting matrix pairs, and that the algebraic set as a scheme is reduced when the base field is of characteristic zero. We also prove that the algebraic set comprising the n × n matrix pairs whose commutators have vanishing diagonal entries is an irreducible complete intersection, and, as a scheme, is reduced when the base field is of characteristic zero.

For massive multiple-input multiple-output (m-MIMO) uplink, the performances of the linear minimum mean-square error (MMSE) detector are considered near optimal, and they occupy benchmark place for most linear iterative detectors. However, the MMSE algorithm is known by its load computational complexity due to the implication of large-scale matrix inversions, and in other hand, iterative methods are often preferred in signal detection because of its low complexity. In this paper, we propose a New Damped Jacobi (NDJ) detector in order to improve the performance of the classical Jacobi linear algorithm. Starting from the classical Jacobi technique to our new proposal, we go through the development of two variants; one uses a damping factor and the other uses a stair-matrix. However, the NDJ incorporates a damping factor in its construction and basing also on stair matrix instead of diagonal matrix. The performances in terms of convergence and low complexity of each Jacobi variant studied in this paper are analyzed. Finally, some simulation examples are given to illustrate the advantages of the new proposed algorithm.

For a simple graph G, the signless Laplacian matrix of G, denoted by Q ( G ) , is defined as D ( G ) + A ( G ) , where A ( G ) and D ( G ) are the adjacency matrix and the diagonal matrix of vertex...

This paper presents factorizations of each discrete sine transform (DST) matrix of types I, II, III, and IV into a product of sparse, diagonal, bidiagonal, and scaled orthogonal matrices. Based on the proposed matrix factorization formulas, reduced multiplication complexity, recursive, and radix-2 DST I-IV algorithms are presented. We will present the lowest multiplication complexity DST-IV algorithm in the literature. The paper fills a gap in the self-recursive, exact, and radix-2 DST I-IIII algorithms executed via diagonal, bidiagonal, scaled orthogonal, and simple matrix factors for any input $n=2^{t} \; (t \geq 1)$ . The paper establishes a novel relationship between DST-II and DST-IV matrices using diagonal and bidiagonal matrices. Similarly, a novel relationship between DST-I and DST-III matrices is proposed using sparse and diagonal matrices. These interweaving relationships among DST matrices enable us to bridge the existing factorizations of the DST matrices with the proposed factorization formulas. We present signal flow graphs to provide a layout for realizing the proposed algorithms in DST-based integrated circuit designs. Additionally, we describe an implementation of algorithms based on the proposed DST-II and DST-III factorizations within a double random phase encoding (DRPE) image encryption scheme.

The error variance-covariance matrices of the monthly GRACE gravity field models are usually well-structured (e.g., order-leading) to contain the error information of the monthly gravity field models, and they are important information to further improve the accuracy of the estimated mass transportation by a post-processing scheme. Intensive studies have been performed to understand the impact of different approximations of the full error variance-covariance matrix on the mass estimates obtained with the conventional GRACE-post processing methods (e.g., the de-striping method). In this study, based on the variants of the mascon approach which treat monthly GRACE solutions as input, we consider four different structures to the error variance-covariance matrix, (i) full matrix, (ii) block diagonal matrix, (iii) diagonal matrix, and (iv) identity matrix, and examine their impact on the mass transport estimates in Greenland. Using both synthetic and real data, we analyze the results at four temporal scales: (i) the long-term decadal scale, (ii) the inter-annual scale, (iii) the seasonal scale, and (iv) the monthly scale. Based on the synthetic study, we find that for the recovery of the long-term trend, the application of the diagonal structure obtains the best estimates. This is caused by the amplification of the model error in the mascon approach when considering the full and block diagonal structures of error variance-covariance matrix, not due to any imperfection of them. Therefore, we emphasize that one should be aware of mascon model deficiency in the mascon approach. Furthermore, the best inter-annual, seasonal and monthly mass estimates are derived by considering the block diagonal and full structures. This is caused by the behavior of the bias and the unique parameterization error in the case of different structures. A similar finding is also presented in the real data case. Finally, our analysis denotes the necessity of releasing the block diagonal structure together with the official monthly gravity field model for the GRACE Follow-On mission, instead of releasing only the diagonal structure as done for the GRACE mission.

In this paper, the problem of fuzzy adaptive control of unknown nonlinear fractional-order systems with external disturbances and unknown control directions is studied. We exploit a decomposition of the control gain matrix into a symmetric positive-definite matrix, a diagonal matrix with diagonal entries + 1 or 1 , and a unity upper triangular matrix. Fuzzy logic systems are used for estimating the unknown nonlinear functions. Based on the fractional Lyapunov direct method and some proposed lemmas, a novel fuzzy adaptive controller is designed. The proposed method can guarantee that all the signals in the closed-loop systems remain bounded and the tracking errors converge to an arbitrary small region of the origin. In addition, for updating the parameters of the fuzzy system, fractional-order adaptations laws are proposed. Lastly, an illustrative example is given to demonstrate the effectiveness of the proposed results.

In this paper, we propose a separation algorithm for weakly correlated source signals. At present, there are many blind source separation algorithms based on independent component analysis, which can only deal with the separation of uncorrelated source signals. However, the assumption that the source signals are independent is too strict in practical application, so the application of the separation algorithm is limited. The proposed weak correlation source signal separation algorithm is divided into two stages. In the first stage, the correlation matrices of mixed signals are first decomposed into two parts, and then their image matrices are introduced. Finally, the objective function is constructed. Using the iterative algorithm to compute this function, we prove that when the function tends to zero, similar matrices of independent parts of the aforementioned correlation matrices are obtained. In the second stage, the joint diagonal matrix model is established by using these similarity matrices, and finally the separable separation matrix is obtained by optimizing the model so as to achieve the purpose of separating the source signals. Simulation experiments also verify the effectiveness of the algorithm.

A sum of a large-dimensional random matrix polynomial and a fixed low-rank matrix polynomial is considered. The main assumption is that the resolvent of the random polynomial converges to some deterministic limit. A formula for the limit of the resolvent of the sum is derived and the eigenvalues are localised. Three instances are considered: a low-rank matrix perturbed by the Wigner matrix, a product $HX$ of a fixed diagonal matrix $H$ and the Wigner matrix $X$ and a special matrix polynomial. The results are illustrated with various examples and numerical simulations., Comment: 32 pages, 6 figures

The singular value decomposition (SVD) based hybrid processing for millimeter wave (mmWave) multiple-input multiple-output (MIMO) systems attempts to eliminate mutual interference between data streams by processing the original channel matrix to an approximate diagonal matrix. However, the non-zero off-diagonal entries will result in performance loss. To improve the performance with low-complexity, we propose a multistage optimal detection algorithm for SVD-based hybrid mmWave MIMO systems. At the first stage, a decision criterion is proposed by effectively exploiting the characteristic that the magnitudes of off-diagonal entries of the baseband channel matrix are relatively small. Through this decision criterion, some elements of global optimal solution can be determined. The ratio of the determined elements to all elements is analyzed to reveal the advantage of proposed decision criterion. At the second stage, we further propose a decision feedback algorithm to detect more elements of global optimal solution. If all elements are determined, the detection is complete, otherwise it enters the third stage. In the case of third stage, we apply the maximum likelihood (ML) detection algorithm to solve the residual small-scale optimization problem containing only few undetermined elements. Numerical experiments show that the number of elements required to be determined by ML detection is relatively very small or zero, in most situations, that is why the algorithm has a significantly lower complexity.

The perturbed GUE corners ensemble is the joint distribution of eigenvalues of all principal submatrices of a matrix $G+\mathrm{diag}(\mathbf{a})$, where $G$ is the random matrix from the Gaussian Unitary Ensemble (GUE), and $\mathrm{diag}(\mathbf{a})$ is a fixed diagonal matrix. We introduce Markov transitions based on exponential jumps of eigenvalues, and show that their successive application is equivalent in distribution to a deterministic shift of the matrix. This result also leads to a new distributional symmetry for a family of reflected Brownian motions with drifts coming from an arithmetic progression. The construction we present may be viewed as a random matrix analogue of the recent results of the first author and Axel Saenz (arXiv:1907.09155 [math.PR])., Comment: 14 pages, 1 figure. v3: minor fixes in proof of Thm. 4.4; typos fixed

For mixed uncertain multisensor networked systems simultaneously with four uncertainties including random one-step measurement delays, missing measurements, multiplicative noises and uncertain noise variances, three new approaches of solving robust fusion estimation problem are presented. They include augmented state approach with fictitious white noises, extended Lyapunov equation approach, and universal integrated covariance intersection (ICI) fusion approach. Applying them, the minimax robust local and five fused time-varying Kalman estimators (predictor, filter an smoother) are presented in the sense that their actual estimation error variances are guaranteed to have the corresponding minimal upper bounds for all admissible uncertainties. The five robust fusers include centralized fuser, fusers weighted respectively by matrices, diagonal matrices and scalars, and ICI fuser. Their robustness and accuracy relations are proved. The proposed approaches and results constitute an important methodology and a unified robust fusion Kalman filtering theory of solving the robust estimation problem. A simulation example applied to the vehicle suspension system shows their effectiveness and applicability.

Let G be a digraph and A ( G ) be the adjacency matrix of G. Let D ( G ) be the diagonal matrix with outdegrees of vertices of G. For any real α ∈ [ 0 , 1 ] , define the matrix A α ( G ) as A α ( G...

This paper considers the three-phase grid-connected inverter (GCI) as a MIMO system, and presents some new perspectives on the stability of GCI. It is found that with traditional Nyquist criterion (NC)-based method which ignores the off-diagonal components of impedance or admittance matrix, the stability results of impedance and admittance models are different. To make up this disadvantage, a simple stability analysis method based on system feedback ratio matrix (SFRM) is proposed. Instead of ignoring the couplings of impedance or admittance, the SFRM in the proposed method can be equivalent to diagonal matrix by adding a perturbation matrix into the system model, so identical results of stability analysis under impedance and admittance models are attained. Furthermore, to ensure the correctness of stability result, the boundary conditions of the proposed method are also provided. Theoretical analysis, simulation and experimental results show that compared with the state of art methods, the proposed SFRM-based method is of low computational complexity and high accuracy, and is more suitable for engineering application.

This paper presents a class of low memory quasi-Newton methods with standard backtracking line search for large-scale unconstrained minimization. The methods are derived by means of least change updating technique analogous to that for the DFP method except that the full quasi-Newton matrix has been replaced by some diagonal matrix. We establish convergence properties for some particular members of the class under line search with Armijo condition. Sufficient conditions for the methods to be superlinearly convergent are also given. Numerical results are then presented to illustrate the usefulness of these methods in large-scale minimization.

We extend the positive-definite and skew-Hermitian splitting (PSS) iterative method to EPSS (for extended PSS) for solving non-Hermitian positive-definite systems of linear equations. In this kind of extension, the shift matrix is replaced by a Hermitian positive-definite matrix Σ . It is proved that the method is unconditionally convergent. Then, the EPSS method with a 2 × 2-block diagonal matrix Σ as the shift matrix is applied to generalized saddle point problems and the convergence of the method is verified. Naturally, the induced preconditioner can be applied to generalized saddle point problems. In some special cases, this preconditioner coincides with some existing preconditioners. A special case of the EPSS preconditioner, say SEPSS, is used to accelerate the convergence of GMRES. Effectiveness of this preconditioner is investigated by some numerical experiments.

Let $G$ be a graph with adjacency matrix $A(G)$ and let $D(G)$ be the diagonal matrix of the degrees of $G$. The $\alpha-$ index of $G$ is the spectral radius $\rho_{\alpha}\left( G\right)$ of the matrix $A_{\alpha}\left( G\right)=\alpha D\left( G\right) +(1-\alpha)A\left( G\right)$ where $\alpha \in [0,1]$. Let $T_{n,k}$ be the tree of order $n$ and $k$ pendent vertices obtained from a star $K_{1,k}$ and $k$ pendent paths of almost equal lengths attached to different pendent vertices of $K_{1,k}$. It is shown that if $\alpha\in\left[ 0,1\right) $ and $T$ is a tree of order $n$ with $k$ pendent vertices then% \[ \rho_{\alpha}(T)\leq\rho_{\alpha}(T_{n,k}), \] with equality holding if and only if $T=T_{n,k}$. This result generalizes a theorem of Wu, Xiao and Hong \cite{WXH05} in which the result is proved for the adjacency matrix ($\alpha=0$). Let $q=[\frac{n-1}{k}]$ and $n-1=kq+r$, $0 \leq r \leq k-1$. It is also obtained that the spectrum of $A_{\alpha}(T_{n,k})$ is the union of the spectra of two special symmetric tridiagonal matrices of order $q$ and $q+1$ when $r=0$ or the union of the spectra of three special symmetric tridiagonal matrices of order $q$, $q+1$ and $2q+2$ when $r \neq 0$. Thus the $\alpha-$ index of $T_{n,k}$ can be computed as the largest eigenvalue of the special symmetric tridiagonal matrix of order $q+1$ if $r=0$ or order $2q+2$ if $r\neq 0$.

This letter proposes a method to obtain integer reversible discrete cosine transforms for generic transform-based coding schemes. The novelty of the proposed method, which is based on decomposition of the DCT-II matrix into two triangular and one diagonal matrices, is twofold: (i) the new matrices can be of arbitrary size, i.e., any square $N\times N$ dimension, thus suitable for applications where non power-of-2 dimensions are required; (ii) they can be designed with adjustable precision in a trade-off with the number of representation bits. Furthermore, improvements are also proposed over the base scheme to avoid numerical issues when working with large matrices and to obtain more reliable approximations. The performance evaluation demonstrate the effectiveness of the proposed transforms to approximate the coding gain capabilities of the original DCT-II.

Multi-sensor distributed fusion estimation algorithms based on machine learning are proposed in this paper. Firstly, using local estimations as inputs and estimations of three classic distributed fusion (weighted by matrices, by diagonal matrices and by scalars) as the training sets, three distributed fusion algorithms based on BP network (BP net-based fusion weighted by matrices, by diagonal matrices and by scalar) are proposed and the selection basis of the number of nodes in hidden layer is given. Furthermore, by using local estimations as inputs and centralized fusion estimation as training set, another recurrent net-based distributed fusion algorithm is proposed, in the case that neither true states nor cross-covariance matrices is available. This method is not limited to the linear minimum variance (LMV) criterion, so its accuracy is higher than the classical three distributed fusion algorithms. A radar tracking simulation verifies the effectiveness of the proposed fusion networks.

The structure of the Hessian matrix obtained from the locational cost used in coverage control is investigated to provide conditions on the optimality of coverage control solutions. It is shown that in arbitrary dimensions, the Hessian matrix is composed of the direct sum of three well-structured matrices: 1) a diagonal matrix; 2) a block-diagonal matrix; and 3) block-Laplacian matrix. This structure is exploited in the one-dimensional case, where an alternative proof of a sufficient condition for optimality is given. A relationship is shown between centroidal Voronoi tessellation (CVT) configurations and the sufficient condition for optimality via the spatial derivative of the density provided in the cost. A decomposition is used to provide insight into the terms which most affect optimality. Several classes of density functions are analyzed under the proposed condition. Experiments on a multi-robot team are shown to verify theoretical results.

For massive multiple-input multiple-output (MIMO) systems, linear precoding is preferable to nonlinear precoding for better performance-complexity trade-off. However, linear precoding is still difficult to implement in practice for such large systems, because the precoding matrix involves the complicated matrix inversion that must be rapidly computed in real-time. In this paper, we use the large-scale property of massive MIMO systems, the excellent characteristics of the weighted Neumann series (WNS) matrix, and steepest descent (SD) method to devise a new iterative precoding for massive MIMO systems. First, by exploiting the characteristics of WNS iteration, we proposed a weighted Neumann series-steepest descent (WNS-SD) iterative algorithm to perform precoding with low-complexity, and the convergence condition is always met in underloaded scenarios. Second, by devising a novel first iterative step of the aforementioned WNS-SD iterative algorithm, we proposed an accelerated iterative algorithm, named the accelerated weighted Neumann series-steepest descent (AWNS-SD) algorithm. Furthermore, in the first iterative step of the AWNS-SD algorithm, we designed a promising preconditioning matrix for the SD algorithm based on the WNS matrix. Subsequently, via merger with the WNS iterative method, the AWNS-SD iterative precoding not only greatly improves the convergence rate of WNS-SD and other competitive precoding algorithms while maintaining low-complexity, but also guarantees a wide range of convergence. Finally, simulation results verify the validity of the theoretical analysis and show that without additional iteration steps, the proposed AWNS-SD precoding achieves near-optimal performance of the exact zero forcing (ZF) precoding in just one iterative step for typical massive MIMO system configurations. Furthermore, the diagonal matrix concept is applied to the preconditioning technique to further reduce the overall complexity.

In this paper we consider distributed adaptive stabilization for uncertain multivariable linear systems with a time-varying diagonal matrix gain. We show that uncertain multivariable linear systems are stabilizable by diagonal matrix high gains if the system matrix is an H-matrix with positive diagonal entries. Based on matrix measure and stability theory for diagonally dominant systems, we consider two classes of uncertain linear systems, and derive a threshold condition to ensure their exponential stability by a monotonically increasing diagonal gain matrix. When each individual gain function in the matrix gain is updated by state-dependent functions using only local state information, the boundedness and convergence of both system states and adaptive matrix gains are guaranteed. We apply the adaptive distributed stabilization approach to adaptive synchronization control for large-scale complex networks consisting of nonlinear node dynamics and time-varying coupling weights. A unified framework for adaptive synchronization is proposed that includes several general design approaches for adaptive coupling weights to guarantee network synchronization., Comment: 16 Pages and 7 figures

We establish first-order Darboux transformations for the two-dimensional Dirac equation with diagonal matrix potential. The potential is allowed to depend arbitrarily on both variables. The systems we consider here include the scenario of a position-dependent mass as well as the massless case. Our Darboux transformations are more general than their existing counterparts (Pozdeeva and Schulze-Halberg in J Math Phys 51:113501, 2010).

Several approximate matrix inversion methods have been used in linear massive MIMO uplink detectors where their convergence rate, performance, and complexity are greatly affected by the initial solution. In this Letter, the authors exploit a stair matrix, instead of a diagonal matrix, in initialising iterative linear minimum mean square error massive MIMO detector based on several approximate matrix inversion methods, namely, the Gauss-Seidel, successive over relaxation, Richardson iteration, and Newton iteration methods. Numerical results show a significant performance enhancement without a burden of extra complexity using a stair matrix over a diagonal matrix in all methods.

In this paper we are exploring the bitwise connection between the nodes of a interconnection network. We are taking PDN as a model, First of all we are converting the interconnection network into its equivalent connectivity matrix .Then Row/Column vectors of connectivity matrix is used to present the value of a particular node of interconnection network which is shown in Figure 1 as state diagram of PDN which is δ=2. Each bitwise vector shows connectivity with another node in position of bit 1.The vectors also shows the mathematical property of PDN, it means the value of vector of a node in the connectivity matrix preserves the mathematical property of the topology. We assume that each node is connected to itself as a self loop in connectivity matrix. Therefore the diagonal matrix is always 1. The presence of 1’s in a vector (Excluding the self loop) shows degree of the node .The connectivity and its complexity will be explored by using logical operators between the nodes of a PDN so that we can develop algorithms for automatic switching between two nodes automatically. In the due course of study we found many patterns of binary/logical relationship between the nodes which will be discussed in our future discussion in this paper.

For a simple graph $G(V,E)$ with $n$ vertices, $m$ edges, vertex set $V(G)=\{v_1, v_2, \dots, v_n\}$ and edge set $E(G)=\{e_1, e_2,\dots, e_m\}$, the adjacency matrix $A=(a_{ij})$ of $G$ is a $(0, 1)$-square matrix of order $n$ whose $(i,j)$-entry is equal to 1 if $v_i$ is adjacent to $v_j$ and equal to 0, otherwise. Let $D(G)={diag}(d_1, d_2, \dots, d_n)$ be the diagonal matrix associated to $G$, where $d_i=\deg(v_i),$ for all $i\in \{1,2,\dots,n\}$. The matrices $L(G)=D(G)-A(G)$ and $Q(G)=D(G)+A(G)$ are respectively called the Laplacian and the signless Laplacian matrices and their spectra (eigenvalues) are respectively called the Laplacian spectrum ($L$-spectrum) and the signless Laplacian spectrum ($Q$-spectrum) of the graph $G$. If $0=\mu_n\leq\mu_{n-1}\leq\cdots\leq\mu_1$ are the Laplacian eigenvalues of $G$, Brouwer conjectured that the sum of $k$ largest Laplacian eigenvalues $S_{k}(G)$ satisfies $S_{k}(G)=\sum\limits_{i=1}^{k}\mu_i\leq m+{k+1 \choose 2}$ and this conjecture is still open. If $q_1,q_2, \dots, q_n$ are the signless Laplacian eigenvalues of $G$, for $1\leq k\leq n$, let $S^{+}_{k}(G)=\sum_{i=1}^{k}q_i$ be the sum of $k$ largest signless Laplacian eigenvalues of $G$. Analogous to Brouwer's conjecture, Ashraf et al. conjectured that $S^{+}_{k}(G)\leq m+{k+1 \choose 2}$, for all $1\leq k\leq n$. This conjecture has been verified in affirmative for some classes of graphs. We obtain the upper bounds for $S^{+}_{k}(G)$ in terms of the clique number $\omega$, the vertex covering number $\tau$ and the diameter of the graph $G$. Finally, we show that the conjecture holds for large families of graphs.

The problem of decomposing a given covariance matrix as the sum of a positive semi-definite matrix of given rank and a positive semi-definite diagonal matrix, is considered. We present a projection-type algorithm to address this problem. This algorithm appears to perform extremely well and is extremely fast even when the given covariance matrix has a very large dimension. The effectiveness of the algorithm is assessed through simulation studies and by applications to three real datasets that are considered as benchmark for the problem. A local convergence analysis of the algorithm is also presented.

In this paper, we construct random two-faced families of matrices with non-Gaussian entries to approximate a bi-free central limit distribution with a positive definite covariance matrix. We prove that, under modest conditions weaker than independence, a family of random two-faced families of matrices with non-Gaussian entries is asymptotically bi-free from a two-faced family of constant diagonal matrices.

We study diagonal matrices which are left comorphic as elements in full matrix rings. In particular, it is proved that an elementary divisor ring R is left comorphic if and only if all full matrix ...

The paper studies two numerical characteristics of matrix incidence algebras over finite fields associated with generating sets of such algebras: the minimal cardinality of a generating set and the length of an algebra. Generating sets are understood in the usual sense, the identity of the algebra being considered a word of length 0 in generators, and also in the strict sense, where this assumption is not used. A criterion for a subset to generate an incidence algebra in the strict sense is obtained. For all matrix incidence algebras, the minimum cardinality of a generating set and a generating set in the strict sense are determined as functions of the field cardinality and the order of the matrices. Some new results on the lengths of such algebras are obtained. In particular, the length of the algebra of “almost” diagonal matrices is determined, and a new upper bound for the length of an arbitrary matrix incidence algebra is obtained.

Using the algebraic techniques of formal series, we obtain a combinatorial decomposition of some matrices generated by the generalized rencontres polynomials. Specifically, for the complete matrix and the Hankel matrix we obtain an LDU-decomposition R D S T , where D is a diagonal matrix and R and S are two Sheffer matrices. Then, for the shifted matrices we obtain an LTU-decomposition R T S T , where T is a tridiagonal matrix and R and S are two Sheffer matrices. Furthermore, we give an algebraic characterization of the generalized rencontres numbers and polynomials in terms of Hankel determinants. Finally, we determine a relation between the complete matrices and the Hankel matrices.

The work is dedicated to the construction of numerical-analytical method of designing efficient algorithms for the solution of problems in economics and engineering. Using a priori information about the smoothness of the solution, great attention is paid to the construction of high-accuracy solutions. The proposed approach eliminates recurrent structure calculations unknown vectors decisions, which leads to the accumulation of rounding errors. Parallel form of the algorithm is the maximum, and therefore has the shortest possible time the implementation on parallel computing systems. Most conventional algorithms for solving these problems (sweep techniques, decomposition of the matrix into a product of two diagonal matrices, doubling, etc.) when multiple processors work typically no faster than if a single processor. The reason for this is substantial sequence computations of these algorithms.