Real Square Research Articles

Euclidean and circum-Euclidean distance matrices: characterizations and interlacing property

Motivated by the inverse formula of the distance matrix of a tree and the Moore–Penrose inverse of a circum-Euclidean distance matrix (CEDM), in this paper, we study a general real square matrix M whose Moore–Penrose inverse can be expressed as the sum of a Laplacian-like matrix L and a rank one matrix. In particular, for a symmetric hollow matrix M, under an assumption, we show that M is a Euclidean distance matrix if and only if L is positive semidefinite. Based on this, we obtain a new characterization for CEDMs involving their Moore–Penrose inverses. As an application, we show that the distance matrices of block graphs and odd-cycle-clique graphs are CEDMs. Finally, we establish an interlacing property between the eigenvalues of a Euclidean distance matrix M (including the singular case) and its associated Laplacian-like matrix L, which generalizes the interlacing property proved for the distance matrices of trees.

On Certain Algebraic Properties of Symbolic 3-Plithogenic Real Square Matrices

The main objective of this article is to study the inverse of invertible symbolic 3-plithogenic real square matrices using the concept of adjoints and characteristic polynomials. Also, the symbolic 3-plithogenic version of Cayley-Hamilton theorem was proved and provided enough examples to enhance understanding.

Dual core-nilpotent decomposition and dual binary relation

In this paper, we present the definitions of appreciable index and dual index of a dual real square matrix. Then, we introduce the unique dual core-nilpotent (D-C-N) decomposition that exists universally for all dual real square matrices. By applying the decomposition, we get a characterization of the dual Drazin generalized inverse (D-Drazin), introduce D-Drazin, D-C-N, C-sharp C-N (C-C-N) and G-sharp C-N (G-C-N) binary relations of dual real matrices, and discuss relevant properties of these binary relations. We prove that D-Drazin binary relation is a pre-order, and D-C-N, C-C-N and G-C-N binary relations are partial orders. Furthermore, we discuss relationships among binary relations mentioned above.

Validation of public stigma of tuberculosis scale during the COVID-19 pandemic using the Rasch model.

Tuberculosis (TB) is still a major global health problem, and it has been particularly concerning during the COVID-19 pandemic. Non-compliance with anti-TB treatment increases the number of multidrug-resistant cases, causing ongoing transmission and increased morbidity and mortality. The main factors causing TB patients' non-compliance are stigma and lack of financial resources. Stigma harms patients and may cause them to delay seeking and adhering to treatment. Thus, it is important to measure the public stigma surrounding TB. However, few scales are available to measure this stigma as it developed during the COVID-19 pandemic. This study, therefore, aimed to develop and validate such a scale. Mixed methods were employed in this study, consisting of a qualitative phase using in-depth interviews with 26 community leaders and a descriptive quantitative survey of 37 people in the Sumedang District to validate the public stigma of tuberculosis scale during the COVID-19 pandemic. The qualitative data were analyzed using thematic analysis, and the quantitative data were analyzed using the Rasch model. The 21 items yielded by an initial qualitative analysis of the data gathered were validated using the RASCH model, yielding 17 valid items with a Cronbach's Alpha of 0.95, person separation of 3.61, real root mean square deviation (RMSE) of 0.37, infit mean square (INFIT MNSQ) of > +1.25, differential item functioning (DIF) of 1.000, the raw variance of 52.4%, and an unexplained variance ranging from 3.4% to 6.9%. The scale developed to measure the public stigma surrounding TB during the COVID-19 pandemic is valid and reliable to measure stigma surrounding TB in the community, especially the pandemic. Further research is needed to apply the scale to bigger and broader populations to evaluate its measurement consistency.

Square matrices with the inverse diagonal property

We identify the class of real square invertible matrices A for which the signs of the diagonal entries of A−1 match those of A, and begin their study. We say such matrices have the inverse diagonal property (IDP). This class includes many important classes: the positive definite matrices, the M-matrices, the totally positive matrices and some variants, the P-matrices, the diagonally dominant and H-matrices and their inverse classes, as well as triangular matrices. This class is closed under any real invertible diagonal multiplication on either the right or the left. So questions about this class can be reduced to the case of positive diagonal entries. Other basic properties are given. One theme is what conditions need be added to the IDP to insure membership in a familiar class. For example, the positive definite matrices are characterized as certain IDP matrices with special conditions on certain particular principal minors. The tridiagonal case is highlighted. Certain specially simple conditions on such matrices are mentioned that ensure them to be P-matrices, positive definite matrices or M-matrices. We also note that recent results about the invertibility of weakly diagonally dominant matrices are used. Examples are given throughout the paper.

On Novel Results about the Algebraic Properties of Symbolic 3-Plithogenic and 4-Plithogenic Real Square Matrices

Symbolic n-plithogenic sets are considered to be modern concepts that carry within their framework both an algebraic and logical structure. The concept of symbolic n-plithogenic algebraic rings is considered to be a novel generalization of classical algebraic rings with many symmetric properties. These structures can be written as linear combinations of many symmetric elements taken from other classical algebraic structures, where the square symbolic k-plithogenic real matrices are square matrices with real symbolic k-plithogenic entries. In this research, we will find easy-to-use algorithms for calculating the determinant of a symbolic 3-plithogenic/4-plithogenic matrix, and for finding its inverse based on its classical components, and even for diagonalizing matrices of these types. On the other hand, we will present a new algorithm for calculating the eigenvalues and eigenvectors associated with matrices of these types. Also, the exponent of symbolic 3-plithogenic and 4-plithogenic real matrices will be presented, with many examples to clarify the novelty of this work.

An Efficient Hardware Implementation for Complex Square Root Calculation Using a PWL Method

In this paper, we propose a methodology for computing the square root of a complex number based on a piecewise linear (PWL) approximation method. The proposed method relies on a software-based segmentor that automatically divides the three real square root functions used in complex square root computation into the fewest segments with a predefined fractional bit width and maximum absolute error (MAE). The coefficients, including the start point, end point, slope and y-intercept of each segment, are stored for use in the implementation of the hardware design. The proposed fully pipelined circuit is coded in the Verilog hardware description language (HDL). The results of synthesis in TSMC (Taiwan Semiconductor Manufacturing Company) 65-nm CMOS technology show that our design achieves savings of 64.21% in area, 16.67% in delay and 65.08% in power compared to the existing methods. Moreover, implementation results on an FPGA (Field-Programmable Gate Array) platform (XC7Z020-CLG400) show that the proposed design reduces the number of LUTs by 29.38%, delay by 28.57% and power consumption by 53.47%.

Lower bounds for the smallest singular values of generalized asymptotic diagonal dominant matrices

This article presents three classes of real square matrices. They are models of coefficient matrices of linearized Galerkin’s equations. These Galerkin’s equations are derived from first order nonlinear delay differential equation with smooth nonlinearity. This paper shows results of computer experiments stating that the minimum singular values of these matrices are unchanged even if orders of matrices are increased. A time variant Hutchinson equation is also considered. A computer experimental result is shown about this equation. This result shows that “the minimum singular values invariant-ness” holds also for the coefficient matrices of linearized Galerkin’s equations of this equation. A remark is presented that this property can be seen experimentally for a wide class of nonlinear delay differential equation with smooth nonlinearity. A theorem is presented based on the Schur complement. Through it, tight lower bounds are derived for the minimum singular values of such three matrices. It is proved that these lower bounds are unchanged even if orders of matrices are increased.

Constructing the field of values of decomposable and general matrices using the ZNN based path following method

AbstractThis paper describes and develops a fast and accurate path following algorithm that computes the field of values boundary curve for every conceivable complex or real square matrix . It relies on the matrix flow decomposition algorithm that finds a proper block‐diagonal flow representation for the associated hermitean matrix flow under unitary similarity if that is possible. Here is the 1‐parameter‐varying linear combination of the real and skew part matrices and of . For indecomposable matrix flows, has just one block and the ZNN based field of values algorithm works with directly. For decomposing flows , the algorithm decomposes the given matrix unitarily into block‐diagonal form with diagonal blocks whose individual sizes add up to the size of . It then computes the field of values boundaries separately for each diagonal block using the path following ZNN eigenvalue method. The convex hull of all sub‐fields of values boundary points finally determines the field of values boundary curve correctly for decomposing matrices . The algorithm removes standard restrictions for path following FoV methods that generally cannot deal with decomposing matrices due to possible eigencurve crossings of . Tests and numerical comparisons are included. Our ZNN based method is coded for sequential and parallel computations and both versions run very accurately and fast when compared with Johnson's Francis QR eigenvalue and Bendixson rectangle based method and compute global eigenanalyses of for large discrete sets of angles more slowly.

Diagonal canonical form of interval matrices and applications on dynamical systems

Finding the simplest form of a set of quantities is an important aspect of any branch of Mathematics. Of course, the simplest form or the canonical form as we often call it in mathematics, must possess all the important characteristics of the set of quantities. A real square matrix satisfying certain conditions can be brought to diagonal form which is its simplest form such that the diagonal form retains the eigenvalues, determinants, trace, rank, nullity,.. of the original matrix. Many computations with matrices become easier if one can diagonalize the matrices. In this article, we suggest an approach for diagonalizing interval matrices employing a novel methodology called the pairing technique, which will make it simpler and more effective to classify and investigate interval matrices. This is extremely useful because computations involving a diagonalizable matrix can often be reduced to much simpler computations involving the corresponding diagonal matrix. We also discuss two real world applications on planar systems and linear discrete dynamical systems.

Core-EP Monotonicity Characterizations for Property-n Matrices

A square matrix is said to have property n if there exists a positive integer w such that Aw is nonnegative. In this paper, we study the core-EP monotonicity for property-n matrices. Some necessary and sufficient conditions for a property-n matrix to be core-EP monotone are given. Moreover, a necessary and sufficient condition for a real square matrix to have a nonnegative core-EP inverse is also presented.

STATISTICAL CONVERGENCE OF MATRIX SEQUENCES

This paper extends the statistical convergence of real or complex numbers to the real square matrices sequences . In this context, we investigate the relation between the statistical convergence, statistical Cauchy condition, strong Ces\`{a}ro summability of matrices sequences. This leads us to an initial analysis of the Tauberian conditions for the statistical convergence of matrix sequences.

Multivariate continuous-time autoregressive moving-average processes on cones

In this article we study multivariate continuous-time autoregressive moving-average (MCARMA) processes with values in convex cones. More specifically, we introduce matrix-valued MCARMA processes with Lévy noise and present necessary and sufficient conditions for processes from this class to be cone valued. We derive specific hands-on conditions in the following two cases: First, for classical MCARMA on Rd with values in the positive orthant Rd+. Second, for MCARMA processes on real square matrices assuming values in the cone of symmetric and positive semi-definite matrices. Both cases are relevant for applications and we give several examples of positivity ensuring parameter specifications. In addition to the above, we discuss the capability of positive semi-definite MCARMA processes to model the spot covariance process in multivariate stochastic volatility models. We justify the relevance of MCARMA based stochastic volatility models by an exemplary analysis of the second order moment structure of positive semi-definite well-balanced Ornstein–Uhlenbeck based models.

A Mathematical Tool to Investigate the Stability Analysis of Structured Uncertain Dynamical Systems with M-Matrices

The μ-value or structured singular value is a prominent mathematical tool to analyze and synthesize both the robustness and performance of time-invariant systems. We establish and analyze new results concerning structured singular values for the Hadamard product of real square M-matrices. The new results are obtained for structured singular values while considering a set of block diagonal uncertainties. The targeted uncertainties are of two types, that is, pure real scalar block uncertainties and real full-block uncertainties. The eigenvalue perturbation result is utilized in order to determine the behavior of the spectrum of perturbed matrices (A∘B)Δ(t) and ((A∘B)TΔ(t)+Δ(t)(A∘B)).

On signed graphs whose spectral radius does not exceed [formula omitted

The Hoffman program with respect to any real or complex square matrix M associated to a graph G stems from Hoffman's pioneering work on the limit points for the spectral radius of adjacency matrices of graphs does not exceed 2+5. A signed graph G˙=(G,σ) is a pair (G,σ), where G=(V(G),E(G)) is a simple graph and σ:E(G)→{+1,−1} is the sign function. In this paper, we study the Hoffman program of signed graphs. Here, all signed graphs whose spectral radius does not exceed 2+5 will be identified.

Real representations of powers of real matrices and its applications

We give real representations of (or ) based on for a real square matrix A, where is the projection to the generalized eigenspace associated with an imaginary eigenvalue μ of A. Our method is based on the spectral decomposition theorem. As applications, we can easily obtain realifications of representations of solutions of inhomogeneous linear difference equations with constant coefficients.

An Approach To Symbolic n-Plithogenic Square Real Matrices For 9≤ n ≤12

The concept of symbolic n-plithogenic algebraic matrices as symmetric structures with n+1 symmetric classical components with the special definition of the multiplication operation. This paper is dedicated to studying the properties of symbolic 10, and 9-plithogenic real square matrices and 11, 12-plithogenic real matrices from algebraic point of view, where algorithms for computing the eigenvalues and determinants will be proved. Also, the inverse of a symbolic n-plithogenic matrix for the special values n=10, n=9, n=11, and n=12 will be presented.

Spectral properties of certain sequences of products of two real matrices

The aim of this paper is to analyze the asymptotic behavior of the eigenvalues and eigenvectors of particular sequences of products involving two square real matrices $A$ and $B$, namely of the form $B^kA$, as $k\rightarrow \infty$. This analysis represents a detailed deepening of a particular case within a general theory on finite families $\mathcal{F} = \{ A_1, \ldots, A_m \}$ of real square matrices already available in the literature. The Bachmann-Landau symbols and related results are largely used and are presented in a systematic way in the final Appendix.

Sign patterns that allow algebraic positivity

A real square matrix is algebraically positive if there exists a real polynomial f such that f(A) is a positive matrix. In this paper, we give a sufficient condition for a sign pattern matrix to allow algebraic positivity, and give some methods to construct higher-order algebraically positive matrices from some lower-order algebraically positive matrices. We also propose two conjectures related to the problem of allowing algebraic positivity.

Classification of Surfaces of Coordinate Finite Type in the Lorentz–Minkowski 3-Space

In this paper, we define surfaces of revolution without parabolic points in three-dimensional Lorentz–Minkowski space. Then, we classify this class of surfaces under the condition ΔIIIx=Ax, where ΔIII is the Laplace operator regarding the third fundamental form, and A is a real square matrix of order 3. We prove that such surfaces are either catenoids or surfaces of Enneper, or pseudo spheres or hyperbolic spaces centered at the origin.