Product Of Matrices Research Articles

A decomposition of the tensor product of matrices

In this paper we decompose (under unitary equivalence) the tensor product A ? A into a direct sum of irreducible matrices, when A is a 3 x 3 matrix.

Stabilization of Periodic Switched k-Valued Logical Networks

The stabilization of periodic switched $k$ -valued logical networks is investigated in this paper, and some new results are presented. The system considered consists of several $k$ valued logical networks and these networks run in a periodic switching law. First, by using the Cheng product of matrices, a periodic switched $k$ -valued logical (control) network is transformed into a discrete dynamic system which is written as an algebraic form. Second, the switching-state space and the switching-input-state space are defined. Then combining with the algebraic form, some necessary and sufficient conditions for the stability and the stabilization are obtained. An algorithm to find the input sequence that stabilizes the system is also provided. Finally, illustrative examples are given to support the proposed new results.

Constrainted least squares solution of Sylvester equation

<abstract> In this paper, we study several constrainted least squares solutions of quaternion Sylvester matrix equation. We first propose a real vector representation of quaternion matrix and study its properties. By using this real vector representation, semi-tensor product of matrices, swap matrix and Moore-Penrose inverse, we derive compatible conditions and the expressions of several constrainted least squares solutions of quaternion Sylvester equation. </abstract>

Some results on block Kronecker and block Hadamard product of matrices

Some results on block Kronecker and block Hadamard product of matrices

Leader–Follower Flocking for Discrete-Time Cucker–Smale Models With Lossy Links and General Weight Functions

In this article, we investigate leader–follower flocking for the Cucker–Smale model with lossy links and general weight functions. Here, the loss phenomenon of the control packages from the controller to the actuator is characterized by a Bernoulli stochastic variable and the edge weights of the interaction network are determined by a general update rule based on the distance between agents. A method based on the products of substochastic matrices is developed to analyze the flocking behavior with lossy links. By means of this method, a sufficient condition which depends on the agents’ initial states, the topology structure, the successful information transmission rate and the weight function is established. Compared with the previous Cucker–Smale flocking results, which are only applicable to positive and decreasing weight functions with specific forms, our result is more general and can be applied to arbitrary positive and decreasing weight functions with nonzero lower bounds. Finally, our result is illustrated through a simulation example.

English

An m × n matrix R with nonnegative entries is called row stochastic if the sum of entries on every row of R is 1. Let Mm,n be the set of all m × n real matrices. For A, B ∈ Mm,n, we say that A is row Hadamard majorized by B (denoted by A ≺ RHB) if there exists an m × n row stochastic matrix R such that A = R ο B, where X ο Y is the Hadamard product (entrywise product) of matrices X, Y ∈ Mm,n. In this paper, we consider the concept of row Hadamard majorization as a relation on Mm,n and characterize the structure of all linear operators T: Mm,n → Mm,n preserving (or strongly preserving) row Hadamard majorization. Also, we find a theoretic graph connection with linear preservers (or strong linear preservers) of row Hadamard majorization, and we give some equivalent conditions for these linear operators on Mn.

Realistic GUT Yukawa couplings from a random clockwork model

We present realistic models of flavor in SU(5) and SO(10) grand unified theories (GUTs). The models are renormalizable and do not require any exotic representations in order to accommodate the necessary GUT breaking effects in the Yukawa couplings. They are based on a simple clockwork Lagrangian whose structure is enforced with just two (one) vectorlike U(1) symmetries in the case of SU(5) and SO(10) respectively. The inter-generational hierarchies arise spontaneously from products of matrices with order one random entries.

Spectra of products of digraphs

A unified approach to the determination of eigenvalues and eigenvectors of specific matrices associated with directed graphs is presented. Matrices studied include the new distance matrix, with natural extensions to the distance Laplacian and distance signless Laplacian, in addition to the new adjacency matrix, with natural extensions to the Laplacian and signless Laplacian. Various sums of Kronecker products of nonnegative matrices are introduced to model the Cartesian and lexicographic products of digraphs. The Jordan canonical form is applied extensively to the analysis of spectra and eigenvectors. The analysis shows that Cartesian products provide a method for building infinite families of transmission regular digraphs with few distinct distance eigenvalues.

Universality of local spectral statistics of products of random matrices

We derive exact analytical expressions for correlation functions of singular values of the product of M Ginibre matrices of size N in the double scaling limit M,N→∞. The singular value statistics is described by a determinantal point process with a kernel that interpolates between Gaussian unitary ensemble statistic and Dirac-delta (picket-fence) statistic. In the thermodynamic limit N→∞, the interpolation parameter is given by the limiting quotient a=N/M. One of our goals is to find an explicit form of the kernel at the hard edge, in the bulk, and at the soft edge for any a. We find that in addition to the standard scaling regimes, there is a transitional regime which interpolates between the hard edge and the bulk. We conjecture that these results are universal, and that they apply to a broad class of products of random matrices from the Gaussian basin of attraction, including correlated matrices. We corroborate this conjecture by numerical simulations. Additionally, we show that the local spectral statistics of the considered random matrix products is identical with the local statistics of Dyson Brownian motion with the initial condition given by equidistant positions, with the crucial difference that this equivalence holds only locally. Finally, we have identified a mesoscopic spectral scale at the soft edge which is crucial for the unfolding of the spectrum.

Linear k-power Preservers on Tensor Products of Matrices

Invariants and the study of the map preserving a certain invariant play vital roles in the study of the theoretical mathematics. The preserver problems are the researches on linear operators that preserve certain invariants between matrix sets. Based on the result of linear $k$-power preservers on general matrix spaces, in terms of the advantages of matrix tensor products which is not limited by the size of matrices as well as the immense actual background, the study of the structure of the linear $k$-power preservers on tensor products of matrices is essential, which is coped with in this paper. That is to characterize a linear map $f:M_{m_{1}\cdots m_{l}}\rightarrow M_{m_{1}\cdots m_{l}}$ satisfying $f(X_{1}\otimes \cdots \otimes X_{l})^{k}=f\left( (X_{1}\otimes \cdots \otimes X_{l})^{k}\right) $ for all $X_{1}\otimes \cdots \otimes X_{l}\in M_{m_{1}\cdots m_{l}}$.

Event-Based Output Regulation of Boolean Control Networks With Time Delay

This brief focuses on the output regulation problem (ORP) of delayed Boolean control networks (DBCNs) under event-triggered control (ETC). Firstly, resorting to the semi-tensor product (STP) of matrices, the DBCNs model is converted into its algebraic form and the considered ORP is transformed into the set stabilization problem. Secondly, by using the matrix partitioning technique, an algorithm is proposed to find the triggering set and design the controller. Furthermore, a necessary and sufficient condition for solving the event-based ORP is provided. Lastly, a biological example is provided to illustrate the feasibility of the main results.

Integrating Tensor Similarity to Enhance Clustering Performance.

The performance of most clustering methods hinges on the used pairwise affinity, which is usually denoted by a similarity matrix. However, the pairwise similarity is notoriously known for its vulnerability of noise contamination or the imbalance in samples or features, and thus hinders accurate clustering. To tackle this issue, we propose to use information among samples to boost the clustering performance. We proved that a simplified similarity for pairs, denoted by a fourth order tensor, equals to the Kronecker product of pairwise similarity matrices under decomposable assumption, or provide complementary information for which the pairwise similarity missed under indecomposable assumption. Then a high order similarity matrix is obtained from the tensor similarity via eigenvalue decomposition. The high order similarity capturing spatial information serves as a robust complement for the pairwise similarity. It is further integrated with the popular pairwise similarity, named by IPS2, to boost the clustering performance. Extensive experiments demonstrated that the proposed IPS2 significantly outperformed previous similarity-based methods on real-world datasets and it was capable of handling the clustering task over under-sampled and noisy datasets.

Semi-blind joint symbols and multipath parameters estimation of MIMO systems using KRST/MKRSM coding

In this paper, we propose a new MIMO communication system in a time-varying multipath environment, using a Khatri-Rao space-time (KRST) coding combined with a multiple Khatri-Rao product of symbol matrices (MKRSM). It is shown the signals received at the receiver form a tensor which satisfies a (M+2)-order nested PARAFAC model, where (M−1) denotes the number of symbol matrices considered for MKRSM coding. Such a generalization of the nested PARAFAC model to (M+N)-order tensors is first studied from a general point of view, with the discussion of some parameter estimation methods depending on the a priori knowledge on the model. Then, a semi-blind receiver composed of three stages, is developed for jointly estimating the transmitted symbols and the multipath parameters.In the first stage, the transmitted symbols and a matrix unfolding of the effective channel including the fading coefficients and the steering matrices, are estimated using closed-form algorithms based on Khatri-Rao factorizations. In the second one, the channel estimation is refined by means of a simplified least-squares algorithm which takes the column orthonormality assumption on the coding matrix into account. In the third one, an alternating least-squares algorithm, combined with a rectification for the Vandermonde factors containing the spatial steering vectors at the transmitter and receiver sides, is applied to estimate the multipath parameters from the estimated channel. A complexity analysis is made for the receivers, and an expected Cramer-Rao bound related to channel estimation is established. Extensive Monte Carlo simulation results show that the semi-blind receiver which combines the channel estimation refinement with the rectification technique exhibit very interesting performance.

Sampled-data general partial synchronization of Boolean control networks

In this manuscript, the general partial synchronization of a type of drive-response Boolean control networks (BCNs) is proposed for the first time and is addressed by sampled-data feedback control (SDFC). Different from all previous synchronization in the field of BCNs, the general partial generalization requires that the total number of the synchronized nodes exactly maintains at a pre-special value after finite-time step, and it removes the requirement on the anticipated synchronized nodes. To begin with, we define the general partial synchronization of derive-response BCNs based on the Hamming distance of two Boolean vectors. Subsequently, by utilizing the semi-tensor product (STP) of matrices, we obtain a series of criteria for BCNs to achieve general partial synchronization by SDFC. Then, the corresponding SDFC strategy is designed, under which the considered BCNs achieve general partial synchronization. Finally, these results are demonstrated by an illustrative numerical example.

Directly obtaining the polarization properties from measured Mueller matrices

In this work, we present a new method that allows directly obtaining the polarization properties from measured macroscopic Mueller matrices without decomposition. The method is based on exploiting the fact that the canonical decomposition of a macroscopic Mueller matrix has only one unique form. Component matrices in canonical decomposition method are interpreted by using the relationships derived with the form of one possible product in polar decomposition. The macroscopic Mueller matrix with elements containing several polarization parameters is generally expressed as a product of the component matrices. Formulas are presented in terms of the elements of a measured macroscopic Mueller matrix for the parameters representing polarization and depolarization properties of a medium. Finally, theoretical validation of the Mueller matrix with known and unknown polarization properties is presented. The accuracy of this method is validated by analyzing the consistency between the results obtained by the proposed method and those obtained by the polar decomposition, in which the values of polarization effects are considered to be accurate with the fact that the correction matrices calculated by pseudopolar decomposition are all approximately to identity matrices. This method has the advantages of directly obtaining the polarization properties of the measured Mueller matrix without decomposition. As an example, the depth-resolved images of the polarization parameters of the biological tissue calculated with the measured Mueller matrix are presented to demonstrate the capability of our method.

Simplification of Shapley value for cooperative games via minimum carrier

Shapley value is one of the most fundamental concepts in cooperative games. This paper investigates the calculation of the Shapley value for cooperative games and establishes a new formula via carrier. Firstly, a necessary and sufficient condition is presented for the verification of carrier, based on which an algorithm is worked out to find the unique minimum carrier. Secondly, by virtue of the properties of minimum carrier, it is proved that the profit allocated to dummy players (players which do not belong to the minimum carrier) is zero, and the profit allocated to players in minimum carrier is only determined by the minimum carrier. Then, a new formula of the Shapley value is presented, which greatly reduces the computational complexity of the original formula, and shows that the Shapley value only depends on the minimum carrier. Finally, based on the semi-tensor product (STP) of matrices, the obtained new formula is converted into an equivalent algebraic form, which makes the new formula convenient for calculation via MATLAB.

Information-Theoretic Limits for the Matrix Tensor Product

This article studies a high-dimensional inference problem involving the matrix tensor product of random matrices. This problem generalizes a number of contemporary data science problems including the spiked matrix models used in sparse principal component analysis and covariance estimation and the stochastic block model used in network analysis. The main results are single-letter formulas (i.e., analytical expressions that can be approximated numerically) for the mutual information and the minimum mean-squared error (MMSE) in the Bayes optimal setting where the distributions of all random quantities are known. We provide non-asymptotic bounds and show that our formulas describe exactly the leading order terms in the mutual information and MMSE in the high-dimensional regime where the number of rows n and number of columns d scale with d = O(n <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">α</sup> ) for some α <; 1/20. On the technical side, this article introduces some new techniques for the analysis of high-dimensional matrix-valued signals. Specific contributions include a novel extension of the adaptive interpolation method that uses order-preserving positive semidefinite interpolation paths, and a variance inequality between the overlap and the free energy that is based on continuous-time I-MMSE relations.

Estimating Nonlinearity Characteristics for Iterative Transformations of a Vector Space

We present theoretical foundations for the matrix-graphic approach (MGA) to the estimation of characteristics of the sets of essential and nonlinear variables of the composition of transformations of an $$n $$ -dimensional vector space over a field. The ternary nonlinearity matrix corresponds to a transformation, where the $$i $$ th row and the $$j $$ th column of the matrix contain $$0 $$ , $$1$$ , or $$2 $$ if and only if the $$j $$ th coordinate function of the transformation depends on the $$i $$ th variable fictitiously, or linearly, or nonlinearly, $$0\leq i,j < n $$ . MGA is based on the inequality according to which the nonlinearity matrix of the product of transformations is at most (the inequality is elementwise) the product of the nonlinearity matrices of the transformations. We define the multiplication for ternary matrices. The properties are studied of the multiplicative monoid of all ternary matrices of order $$n$$ without zero rows and columns and of the monoid $$\mathbf{\Gamma}_n$$ bijectively corresponding to it of all $$n$$ -vertex digraphs with edges labeled with $$0$$ , $$1 $$ , and $$2 $$ , where each vertex has nonzero indegree and outdegree. The iteration depth (number of multipliers) for transformations is estimated with the use of MGA in which the four types of the nonlinearity of transformations can be achieved, where each or some of the coordinate functions of the product of transformations can depend nonlinearly on all or at least some variables. We present the results of research on the nonlinearity of iterations of round substitution of the block ciphers DES and “Magma.”

Cooperative Pursuit Control for Multiple Underactuated Underwater Vehicles with Time Delay in Three-Dimensional Space

SUMMARYIn this paper, the k-valued logic control network is introduced to study the cooperative pursuit control problem of multiple underactuated underwater vehicles (UUVs) with time delay in three-dimensional space. The semi-tensor product of matrices is used to solve the complex calculation problem of the large dimension matrix. The influence of communication delay on multiple UUVs’ optimization and cooperative pursuit control is expressed in a matrix. Under the leadership of evader UUV, the control algorithm can ensure that all the pursuit UUVs reach the desired position. The stability of the closed loop system is proved.

Analysis of wave propagation through functionally graded porous cylindrical structures considering the transfer matrix method

This paper presents a study on acoustic wave transmission through functionally graded porous (FGP) cylindrical shells. The mechanical properties of the FGP material, such as porosity, vary continuously along the thickness of the cylinder, considering a power-law distribution profile. According to a laminate model, the structure is assumed to be composed of a finite number of isotropic homogeneous porous sublayers. The extended full method (EFM) is used to describe the displacement and stress variables of each porous sublayer. Next, using the transfer matrix method (TMM), a local transfer matrix is established for each sublayer, which connects the field variables of the inner and outer surfaces of the sublayer. Eventually, the product of these transfer matrices for adjacent sublayers forms the global transfer matrix of the structure. The effects of different structural and environmental properties on sound transmission loss (TL) of the system are studied. The results show that TL is enhanced by increasing porosity of the outer surface of the cylinder, although enhancing the inner porosity adversely affects TL. Finally, it has been proved that by functionally grading a porous cylinder, sound insulation of the structure is improved dramatically in both mass and stiffness-controlled regions.