Matrices Of Order Research Articles

The optimization algorithm for the directions of influence of risk factors on the system that manages the potential of machine-building enterprises

The lines of influence of risk factors on the system of management of financial potential of machine-building enterprises were determined. Their application makes it possible to provide enterprises with complete and reliable information on the system of the analysis component groups necessary for making optimal management decisions, improve the system of production and technological management through the use of integrated methods focused on analysis of the entire spectrum of critical parameters of the enterprise activities and maintain its financial potential at a proper functional level. Consistency comparison of expert’s local priorities with regard to the elements of the second and third levels of the hierarchical model of choice of the neutralization method has been made. The consistency relation showing the degree of violation of numerical value of transitive consistency is the criterion for assessing local expert priorities. A random coherence of local priorities of matrices of various orders was obtained. The algorithm of the strategy of optimization of the financial potential of machine-building enterprises was proposed taking into account influence of the risk factors. It is based on strategic tasks and principles and provides a tactical system set of methods, tools and concrete measures to implement the policy of optimizing the financial potential. It is characterized by various parametric indicators that take into account the risk-oriented optimization of enterprise’s financial potential. Depending on situation, the strategy should be adjustable and flexible for implementation in order to obtain the most positive result.

Some spectral norm inequalities on Hadamard products of nonnegative matrices

Let A and B be nonnegative square matrices of the same order. Denote by ‖⋅‖ and ρ(⋅) the spectral norm and the spectral radius respectively. We prove the following inequalities:‖A∘B‖≤‖A∘A‖12‖B∘B‖12;‖A∘B‖≤ρ12(ATB∘BTA)≤ρ12(ATB∘ATB)≤ρ(ATB), where ∘ denotes the Hadamard product. This interpolates the inequality‖A∘B‖≤ρ(ATB) due to Huang. Some spectral norm inequalities for an arbitrarily finite number of nonnegative square matrices are also obtained, which refine some other results of Huang.

Infinite family of equi-isoclinic planes in Euclidean odd dimensional spaces and of complex symmetric conference matrices of odd orders

A n-set of equi-isoclinic planes in Rr is a set of n planes spanning Rr each pair of which has the same non-zero angle arccos⁡λ. We prove that for any odd integer k≥3 such that 2k=pα+1, p an odd prime, α a positive integer the maximum number of equi-isoclinic planes with angle arccos⁡12k−2 in R2k−1 is equal to 2k−1. It is shown that the solution of this geometric problem is obtained by the construction of complex symmetric conference matrices of order 2k−1, and that all these constructions are performed by use of the Legendre symbol of the Galois field GF(pα).

An Approach to Bounding the Spectral Radius of a Weighted Digraph

The paper suggests a general approach to deriving upper bounds for the spectral radii of weighted digraphs. The approach is based on the generalized Wielandt lemma (GWL), which reduces the problem of bounding the spectral radius of a given block matrix to bounding the Perron root of the matrix composed of the norms of the blocks. In the case of the adjacency matrix of weighted graphs and digraphs where all the blocks are square positive (semi)definite matrices of the same order, the GWL takes an especially nice simple form. The second component of the approach consists in applying known upper bounds for the Perron root of a nonnegative matrix. It is shown that the approach suggested covers, in particular, the known upper bounds of the spectral radius and allows one to describe the equality cases.

TWO-CIRCULANT HADAMARD MATRICES, WEIGHING MATRICES, AND RYSER'S CONJECTURE

Introduction: Hadamard matrices and weighing matrices share the same family. The latter can fill up voids in the matrix space by setting some elements to zero, but this feature has not been properly studied yet. Purpose: To study how the orders of orthogonal matrices used in information processing can affect their structure. Results: For Ryser's conjecture about orders critical for cyclic Hadamard matrices, an extension has been suggested, covering Hadamard matrices and weighing matrices which consist of two cyclic blocks. We give examples of Hadamard matrices extended to the newly revealed critical order equal to 32, with symmetrical blocks or, on higher orders, with unsymmetrical blocks. We also present two-circulant weighing matrices which replace Hadamard matrices and alternate with them. There is an exceptional case related to the order 24 on which two-circulant Hadamard matrices or weighing matrices do not exist, forcing you to search for a solution among four-block constructions. A special set of Hadamard matrices of 20and 52-fold orders is pointed out, as their blocks are asymmetric. A new assumption about the critical order 64 is discussed.

Distances on the Commuting Graph of the Ring of Real Martices

The vertices of the commuting graph of a semigroup S are the noncentral elements of this semigroup, and its edges join all pairs of elements g, h that satisfy the relation gh = hg. The paper presents a proof of the fact that the diameter of the commuting graph of the semigroup of real matrices of order n ≥ 3 is equal to 4. A survey of results in that subject matter is presented, and several open problems are formulated.

An interval extension of SMS method for computing weighted Moore–Penrose inverse

An interval extension of successive matrix squaring (SMS) method for computing the weighted Moore---Penrose inverse $$A^{\dagger }_{MN}$$AMN? along with its rigorous error bounds is proposed for given full rank $$m \times n$$m?n complex matrices A, where M and N be two Hermitian positive definite matrices of orders m and n, respectively. Starting with a suitably chosen complex interval matrix containing $$A^{\dagger }_{MN}$$AMN?, this method generates a sequence of complex interval matrices each enclosing $$A^{\dagger }_{MN}$$AMN? and converging to it. A new method is developed for constructing initial complex interval matrix containing $$A^{\dagger }_{MN}$$AMN?. Convergence theorems are established. The R-order convergence is shown to be equal to at least l, where $$l \ge 2$$l?2. A number of numerical examples are worked out to demonstrate its efficiency and effectiveness. Graphs are plotted to show variations of the number of iterations and computational times compared to matrix dimensions. It is observed that ISMS is more stable compared to SMS.

Determinants of Generalized Matrices of Order 2

The concept of the determinant of a generalized matrix of order 2 is introduced, and some properties of such determinants are presented.

Certain Results on Bicomplex Matrices

This paper begins the study of bicomplex matrices. In this paper, we have defined bicomplex matrices, determinant of a bicomplex square matrix and singular and non-singular matrices in C2. We have proved that the set of all bicomplex square matrices of order n is an algebra.We have given some definitions and results regarding adjoint and inverse of a matrix in C2. We have defined three types of conjugates and three types of tranjugates of a bicomplex matrix. With the help of these conjugates and tranjugates, we have also definedsym metric and skew - symmetric matrices, Hermitian and Skew - Hermitian matrices in C2.

Block-pulse functions method for solving three-dimensional fractional Poisson type equations with Neumann boundary conditions

In this paper, a numerical scheme based on the three-dimensional block-pulse functions is proposed to solve the three-dimensional fractional Poisson type equations with Neumann boundary conditions. The differential operational matrices of fractional order of the three-dimensional block-pulse functions are derived from one-dimensional block-pulse functions, which are used to reduce the original problem to solve a system of linear algebra equations. In addition, the convergence analysis of the proposed method is deeply investigated. Lastly, several numerical examples are presented and the numerical results obtained show that our method is effective and feasible.

Memory-usage advantageous block recursive matrix inverse

The inversion of extremely high order matrices has been a challenging task because of the limited processing and memory capacity of conventional computers. In a scenario in which the data does not fit in memory, it is worth to consider exchanging less memory usage for more processing time in order to enable the computation of the inverse which otherwise would be prohibitive. We propose a new algorithm to compute the inverse of block partitioned matrices with a reduced memory footprint. The algorithm works recursively to invert one block of a k × k block matrix M, with k ≥ 2, based on the successive splitting of M. It computes one block of the inverse at a time, in order to limit memory usage during the entire processing. Experimental results show that, despite increasing computational complexity, matrices that otherwise would exceed the memory-usage limit can be inverted using this technique.

Spectral inclusion for unbounded diagonally dominant $n\times n$ operator matrices

In this paper, we establish an analytic enclosure for the spectrum of unbounded linear operators~$\mathcal{A} $ admitting an $n \times n$ matrix representation in a Hilbert space $\mathcal{H} =\mathcal{H} _1\oplus \cdots \oplus \mathcal{H} _n$. For diagonally dominant operator matrices of order 0, we show that this new enclosing set, the block numerical range $W^n(\mathcal{A} )$, contains the eigenvalues of $\mathcal{A} $ and that the approximate point spectrum of $\mathcal{A} $ is contained in its closure $\overline {W^n(\mathcal{A} )}$. Since the block numerical range turns out to be a subset of the usual numerical range, $W^n(\mathcal{A} )\subset W(\mathcal{A} )$, it may give a tighter enclosure of the spectrum. Moreover, we prove Gershgorin theorems for diagonally dominant $n \times n$ operator matrices and compare our results to both Gershgorin bounds and classical perturbation theory. Our results are illustrated by deriving new lower bounds for $3\times 3$ self-adjoint operator matrices and applying the latter to three-channel Hamiltonians in quantum~mechanics.

A scalable iterative dense linear system solver for multiple right-hand sides in data analytics

Abstract We describe Parallel-Projection Block Conjugate Gradient ( pp-bcg ), a distributed iterative solver for the solution of dense and symmetric positive definite linear systems with multiple right-hand sides. In particular, we focus on linear systems appearing in the context of stochastic estimation of the diagonal of the matrix inverse in Uncertainty Quantification. pp-bcg is based on the block Conjugate Gradient algorithm combined with Galerkin projections to accelerate the convergence rate of the solution process of the linear systems. Numerical experiments on massively parallel architectures illustrate the performance of the proposed scheme in terms of efficiency and convergence rate, as well as its effectiveness relative to the (block) Conjugate Gradient and the Cholesky-based ScaLAPACK solver. In particular, on a 4 rack BG/Q with up to 65,536 processor cores using dense matrices of order as high as 524,288 and 800 right-hand sides, pp-bcg can be 2x-3x faster than the aforementioned techniques.

Simple right-alternative unital superalgebras over an algebra of matrices of order 2

Simple right-alternative unital superalgebras over an algebra of matrices of order 2

Linear representation of a graph

In this paper the linear representation of a graph is defined. A linear representation of a graph is a subgroup of $GL(p,\mathbb{R})$, the group of invertible matrices of order $ p $ and real coefficients. It will be demonstrated that every graph admits a linear representation. In this paper, simple and finite graphs will be used, framed in the graphs theory's area

Supraposinormality and hyponormality for the generalized Cesàro matrices of order two

Supraposinormality and hyponormality for the generalized Cesàro matrices of order two

Restricted Constraints in a Max-Flow and Min-Cost problem

The present paper defines a combination of two most important and old flow problems such as maximum flow and minimum cost flow. In the first one a flow with the maximum value from source node to sink node is sought. The second one seeks a flow with the minimum total transferring costs from source nodes to demand nodes. Here an attempt is made to obtain an optimal route of a more realistic situation as to scheduling some restricted constraint, in which its transferring cost is minimized and its value is maximized. The proposed algorithm is formulated and solved by the lexicographic search approach. It is seen that the time required for the search of the optimal solution is fairly less. Also the algorithm is applied to different order matrices with number of stations = 5, 8, 10, …, 50 and in the dimensions of 6 to exhibit its effectiveness.

Use of symmetric Hadamard and Mersenne matrices in digital image processing

Matrices with orthogonal columns (rows) are extensively used in data processing. One of the pressing challenges for developers of image processing systems is the ability to choose easily the optimal types of structured orthogonal matrices for their specific tasks. The purpose of this study is to systemize the main types of structured symmetric Hadamard matrices, such as the cyclic, negacyclic, bicyclic, four-block and Propus-form three-block ones, which can be used in image processing for compression, filtering, antinoise coding, and masking. The basis of Hadamard matrices is extended with quasi-orthogonal Mersenne matrices of odd orders with previously unknown symmetry properties. It was found that Mersenne matrices and Hadamard matrices built using Mersenne matrices are directly associated with numeric sequences, for some of which the cyclic, bicyclic matrices, and Propus-shaped matrices exist. The paper demonstrates that the use of Mersenne matrices as a "core" produces Haramard matrices with new symmetric structures. This fact expands the classification of orthogonal matrices. Images of previously unknown symmetric Hadamard matrices are provided. The practical value of the paper is that the obtained results provide wide opportunities to facilitate the selection of matrices, which best fit for processing of certain images, including images of non-standard sizes. The number of matrix fragments sufficient for reconstruction of the whole matrix is defined for all matrices under consideration. This way of representation of Hadamard symmetric matrices allows us to increase the efficiency of the image processing using lesser the computer memory.

Order statistics of the moduli of the eigenvalues of product random matrices from polynomial ensembles

Let $X_{1}, \ldots , X_{m_{N}}$ be independent random matrices of order $N$ drawn from the polynomial ensembles of derivative type. For any fixed $n$, we consider the limiting distribution of the $n$th largest modulus of the eigenvalues of $X = \prod _{k=1}^{m_{N}}X_{k}$ as $N \to \infty $ where $m_{N}/N$ converges to some constant $\tau \in [0, \infty )$. In particular, we find that the limiting distributions of spectral radii behave like that of products of independent complex Ginibre matrices.

Optical Characterization of a Thin-Film Material Based on Light Intensity Measurements

Light interacts with materials in a variety of ways; this article focuses on determination of refraction and absorption characterized by a material’s refractive index. We discuss some of the useful models for the frequency dependence of the refractive index, and practical approaches to calculating refractive indices of thin films and thick substrates. The efficiency of manufacturing of existing and successful creation of new devices of solid-state micro- and nanoelectronics largely depends on the level of development of the technology for manufacturing layers of various materials with a thickness of several nanometers to tens of micrometers. A high degree of perfection of layered structures and particularly structures based on dielectric and/or metallic films with nanometer thickness is needed for their successful application in micro-, nano-, acousto-, microwave and optoelectronics. It is impossible to achieve high degree of perfection without the use of high-precision methods of measuring electrophysical parameters of dielectric and semiconductor materials and structures, metallic films. We have developed the program “Multilayer”, which serves both to simulate the propagation of light through multilayer thin-film layered media, and to determine the dielectric (permittivity tensor of anisotropic films) and geometric (physical and optical thicknesses of the film) parameters of various thin-film coatings. The base mathematical models applied for the description of the light wave propagation through a homogeneous optical medium and for the determination of the optical characteristics of thin layers of optical materials based on the results of light intensity measurements are described. The main mathematical formalism employed in the program is based on solving the Maxwell’s equations for propagation of light through anisotropic stratified media. The algorithm uses the Berreman matrices of order