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Showing papers on "Square matrix published in 2021"


Journal ArticleDOI
TL;DR: In this article, Nguyen and Vu showed an approximate Spielman-Teng theorem for {0, 1}-valued matrices, each of whose rows is an independent vector with exactly n/2 zero components.
Abstract: An approximate Spielman-Teng theorem for the least singular value sn(Mn) of a random n × n square matrix Mn is a statement of the following form: there exist constants C, c > 0 such that for all η > 0, Pr(sn(Mn) ≤ η) ≲ nCη + exp(−nc). The goal of this paper is to develop a simple and novel framework for proving such results for discrete random matrices. As an application, we prove an approximate Spielman-Teng theorem for {0, 1}-valued matrices, each of whose rows is an independent vector with exactly n/2 zero components. This improves on previous work of Nguyen and Vu, and is the first such result in a ‘truly combinatorial’ setting.

15 citations


Journal ArticleDOI
TL;DR: By combining eigenvectors of the covariance matrix and random normal distribution, a new method has been presented to solve optimization problems and the results have been compared with previous studies.
Abstract: In this paper, by combining eigenvectors of the covariance matrix and random normal distribution, a new method has been presented to solve optimization problems. This method has been inspired by CMA-ES method and has been named eigenvectors of covariance matrix (ECM). ECM generates some solutions in each stage; then it assigns a value to all solutions by utilizing a dynamic penalty function. The best solution in each stage is selected as the answer of optimization problem and ECM tries to push towards a better point. According to the penalty function, the solutions with no violation along with the solutions with little violation are considered as good solutions and named desirable data. By utilizing desirable data, a square matrix is formed and its covariance matrix can be calculated. The eigenvectors of the covariance matrix are orthogonal and they show directions of distribution of desirable points that have been chosen before. We can hopefully get a better answer by moving from the best current answer towards the points which are distributed normally and randomly around the weighted combination of eigenvectors. To ensure the validity and accuracy of ECM method, weights of six truss structures have been optimized through this method and the results have been compared with previous studies.

11 citations


Journal ArticleDOI
TL;DR: In this article, the fundamental fractional exponential matrix (FFEM) is used to solve the problem of linear fractional electrical circuits (ECs) in 2 and 3-dimensional domains.
Abstract: It is well known that the fundamental fractional exponential matrix (FFEM) is closely related to the formal solution of the homogeneous and non-homogeneous linear time-conformable fractional dynamical differential system (LT-CFDDS) with delay in control and it also plays a central role in the solution of any other (matrix) fractional differential system (FDS). In this paper, we present several new theoretical results and computational formulae of the FFEM e A t α α for any arbitrary square matrix A . Moreover, some attractive and interesting special cases for FFEM are also derived and discussed. In addition, three important and interesting physical applications related to linear fractional electrical circuits (ECs) in 2 and 3-dimentions are considered and solved by means of FFEM. These applications are the fractional RC, RL and RLC-ECs. The method exists by converting the data of a given linear fractional EC into homogeneous or non-homogeneous LT-CFDDS for which solutions can be easily computed. For further and simplicity analysis, we provide an illustrative example for each problem and present the general exact solution (GES) in a simple form based on our new approach. Moreover, graphical results are created and discussed in order to ensure that the solutions of specific problems are stable at different values of α and assure that our suggestion technique is a simple, efficient, accurate, powerful analytic tool and can be applied successfully to solve many other conformable fractional differential problems in various fields. Finally, the classical behaviors of physical problems are recovered when the fractional order α is equal to 1.

10 citations


Journal ArticleDOI
TL;DR: In this paper, the authors compare some properties of doubly stochastic operators in finite and infinite dimensions and provide relevant applications of these operators in the existence of solutions for some infinite linear equations and functional equations.

8 citations


Journal ArticleDOI
TL;DR: Weak core matrices (or weak core inverse) as discussed by the authors are a generalization of the core inverse of complex square matrices, and they form a more general class than that given by weak group matrices.
Abstract: In this paper, we introduce a new generalized inverse, called weak core inverse (or, in short, WC inverse) of a complex square matrix. This new inverse extends the notion of the core inverse defined by Baksalary and Trenkler (Linear Multilinear Algebra 58(6):681–697, 2010). We investigate characterizations, representations, and properties for this generalized inverse. In addition, we introduce weak core matrices (or, in short, WC matrices) and we show that these matrices form a more general class than that given by the known weak group matrices, recently investigated by H. Wang and X. Liu.

8 citations


Journal ArticleDOI
TL;DR: Considering the need for a high-power light source to excite the nonlinearity of an optical material, how to reduce the power consumption of the system by quantifying the output of each layer after the softmax operation as an 8-bit value and loading these values into amplitude-only spatial light modulators (SLMs).
Abstract: To take full advantage of the application of neural networks to optical systems, we design an optical neural network based on the principle of free-space optical convolution. In this article, considering the need for a high-power light source to excite the nonlinearity of an optical material, we describe how to reduce the power consumption of the system by quantifying the output of each layer after the softmax operation as an 8-bit value and loading these values into amplitude-only spatial light modulators (SLMs). In addition, we describe how to load the matrix with positive and negative values in the amplitude-only SLM by utilizing Fourier properties of the odd-order square matrix. We apply our six-layer optical network to the classification of Mixed National Institute of Standards and Technology database (MNIST) and Fashion-MNIST and find that the accuracy reaches 92.51% and 80.67%, respectively. Finally, we consider the error analysis, power consumption, and response time of our framework.

8 citations


Journal ArticleDOI
TL;DR: Polya ensembles as mentioned in this paper have nice closure properties under the multiplicative convolution for the first class and under the additive convolutions for the other classes, and they have general identities for group integrals similar to the Harish-Chandra-Itzykson-Zuber integral.
Abstract: We study several kinds of polynomial ensembles of derivative type which we propose to call Polya ensembles. These ensembles are defined on the spaces of complex square, complex rectangular, Hermitian, Hermitian antisymmetric and Hermitian anti-self-dual matrices, and they have nice closure properties under the multiplicative convolution for the first class and under the additive convolution for the other classes. The cases of complex square matrices and Hermitian matrices were already studied in former works. One of our goals is to unify and generalize the ideas to the other classes of matrices. Here, we consider convolutions within the same class of Polya ensembles as well as convolutions with the more general class of polynomial ensembles. Moreover, we derive some general identities for group integrals similar to the Harish–Chandra–Itzykson–Zuber integral, and we relate Polya ensembles to Polya frequency functions. For illustration, we give a number of explicit examples for our results.

7 citations


Proceedings ArticleDOI
02 Apr 2021
TL;DR: The unimodular matrix generated by the proposed algorithm guarantees a key in the Hill Cipher for all matrix sizes (n > 4) and only need two parameters (password 1 and password 2).
Abstract: Hill Cipher is one of the methods used in Cryptography. In the Hill Cipher algorithm, the key square matrix must have an inverse modulo. One special matrix that definitely has an inverse is the unimodular matrix. The unimodular matrix can be used as a key in an encryption process. The purpose of this research is to show an alternative in securing digital image data. The type of cryptography used is symmetric cryptography. The algorithm presented by generating a unimodular matrix using the Logistic Map. First, to get a unimodular matrix, we use an identity matrix. The sequence of real number in (0,1) of Logistic Map then converted into integer number from 0 to 255. That number then occupies upper triangular entries of the unimodular matrix. We use the Elementary Row Operation to obtain the complete matrix as a key. Then the multiplication of matrix modulo is used as an encryption process. The unimodular matrix generated by the proposed algorithm guarantees a key in the Hill Cipher for all matrix sizes (n > 4) and only need two parameters (password 1 and password 2). We also tested the algorithm in python language on two digital images (grayscale and color) with different sizes. The results show that the encrypted images are very difficult to read by third parties. The time needed is based on password 1, that is the size of the key matrix. So, the unimodular matrix and logistic map work very well with Hill Cipher to encrypt a digital image.

6 citations


Journal ArticleDOI
TL;DR: This paper describes an algorithm which computes the characteristic polynomial of a matrix over a field within the same asymptotic complexity, up to constant factors, as the multiplication of two square matrices.

6 citations


Journal ArticleDOI
TL;DR: In this paper, it was shown that every matrix over a prime field is the sum of two tripotents if and only if Char (F ) ≤ 5 and F ≤ 5.

5 citations


Journal ArticleDOI
TL;DR: It is shown that characteristic patterns emerge from plots of the ESPs of the matrices representing the studied squares, and these findings may help to open a new perspective regarding magic squares unsolved problems.

Journal ArticleDOI
01 Jan 2021
TL;DR: In this paper, the authors discuss different characteristics of the BT-inverse of a square matrix introduced by Baksalary and Trenkler, and investigate the relationships between BTinverse and other generalized inverses by Core-EP decomposition.
Abstract: In this paper, we discuss different characteristics of the BT-inverse of a square matrix introduced by Baksalary and Trenkler [On a generalized core inverse, Appl. Math. Comput., 236 (2014), 450-457]. While the BT-inverse is defined by a expression, we present some necessary and sufficient conditions for a matrix to be the BT-inverse. Then we give a canonical form of BT-inverse and investigate the relationships between BT-inverse and other generalized inverses by Core-EP decomposition. Some properties of BT-inverse concerned with some classes of special matrix are identified by Core-EP decomposition. Furthermore new representations of BT-inverse are given by the maximal classes of matrices.

Journal ArticleDOI
TL;DR: In this paper, it was shown that every infinite upper triangular matrix over an arbitrary field has a generalized Jordan normal form, and that any square matrix over a closed field has the same Jordan norm.
Abstract: Any square matrix over an algebraically closed field has a Jordan normal form. In this paper, we prove that every infinite upper triangular matrix over an arbitrary field has a generalized infinite...

Journal ArticleDOI
TL;DR: For a large range of matrix sizes in the domain of interest, this work achieves at least 2/3 of the roofline performance and often substantially outperform state-of-the art CUBLAS results on an NVIDIA Volta GPGPU.
Abstract: General matrix-matrix multiplications with double-precision real and complex entries (DGEMM and ZGEMM) in vendor-supplied BLAS libraries are best optimized for square matrices but often show bad pe...

Journal ArticleDOI
TL;DR: In this paper, the authors construct a linear map by taking two groups of mutually unbiased bases (MUBs) of two Hilbert spaces, and obtain a positive map between two different linear spaces of complex matrices.
Abstract: We study entanglement witnesses (EWs) and construct a linear map by taking two groups of mutually unbiased bases (MUBs) of two Hilbert spaces. We provide two new relations about matrices (may not be square matrices) with properties similar to unitary matrix. Using these two new relations, we obtain a positive map between two different linear spaces of complex matrices. Finally, we calculate a special case by applying our map to get an EW on $\mathbb {C}^{3}\otimes \mathbb {C}^{4}$ . We also find three entangled states detected by this EW.

Journal ArticleDOI
TL;DR: A new family of fixed point iterations that include the classical iterations for the numerical computation of G, fundamental in the analysis of M/G/1-type Markov chains, is introduced and it is proved that the iterations in the new class converge faster than the Classical iterations.
Abstract: We consider the problem of computing the minimal nonnegative solution $G$ of the nonlinear matrix equation $X=\sum_{i=-1}^\infty A_iX^{i+1}$ where $A_i$, for $i\ge -1$, are nonnegative square matrices such that $\sum_{i=-1}^\infty A_i$ is stochastic. This equation is fundamental in the analysis of M/G/1-type Markov chains, since the matrix $G$ provides probabilistic measures of interest. A new family of fixed point iterations for the numerical computation of $G$, that includes the classical iterations, is introduced. A detailed convergence analysis proves that the iterations in the new class converge faster than the classical iterations. Numerical experiments confirm the effectiveness of our extension.

Journal ArticleDOI
TL;DR: In this paper, the authors used the Cauchy-Schwarz inequality to find the vacuum stability conditions for the left-right symmetric potential of a bidoublet and left and right Higgs doublets.
Abstract: The orbit space for a scalar field in a complex square matrix representation obtains a Minkowski space structure from the Cauchy–Schwarz inequality. It can be used to find vacuum stability conditions and minima of the scalar potential. The method is suitable for fields such as a bidoublet, an SU(2) triplet or SU(3) octet. We use the formalism to find the vacuum stability conditions for the left-right symmetric potential of a bidoublet and left and right Higgs doublets.

Journal ArticleDOI
TL;DR: In this paper, the authors give various characterizations of random walks with possibly different steps that have relatively large discrepancy from the uniform distribution modulo a prime p, and use these results to study the distribution of the rank of random matrices over F_p and the equi-distribution behavior of normal vectors of random hyperplanes.
Abstract: In this note we give various characterizations of random walks with possibly different steps that have relatively large discrepancy from the uniform distribution modulo a prime p, and use these results to study the distribution of the rank of random matrices over F_p and the equi-distribution behavior of normal vectors of random hyperplanes. We also study the probability that a random square matrix is eigenvalue-free, or when its characteristic polynomial is divisible by a given irreducible polynomial in the limit n to infinity in F_p. We show that these statistics are universal, extending results of Stong and Neumann-Praeger beyond the uniform model.

Journal ArticleDOI
TL;DR: In this article, the authors consider m independent random rectangular matrices whose entries are independent and identically distributed standard complex Gaussian random variables and assume the product of the m rectangular matrix is an n by n square matrix.

Journal ArticleDOI
TL;DR: In this article, the authors define the notion of Pade approximation of Weyl-Stiltjes transforms on an arbitrary compact Riemann surface of higher genus and explore its connection to integrable systems.
Abstract: The paper has two relatively distinct but connected goals; the first is to define the notion of Pade approximation of Weyl–Stiltjes transforms on an arbitrary compact Riemann surface of higher genus. The data consists of a contour in the Riemann surface and a measure on it, together with the additional datum of a local coordinate near a point and a divisor of degree g. The denominators of the resulting Pade-like approximation also satisfy an orthogonality relation and are sections of appropriate line bundles. A Riemann–Hilbert problem for a square matrix of rank two is shown to characterize these orthogonal sections, in a similar fashion to the ordinary orthogonal polynomial case. The second part extends this idea to explore its connection to integrable systems. The same data can be used to define a pairing between two sequences of line bundles. The locus in the deformation space where the pairing becomes degenerate for fixed degree coincides with the zeros of a “tau” function. We show how this tau function satisfies the Kadomtsev–Petviashvili hierarchy with respect to either deformation parameters, and a certain modification of the 2-Toda hierarchy when considering the whole sequence of tau functions. We also show how this construction is related to the Krichever construction of algebro-geometric solutions.

Journal ArticleDOI
TL;DR: In this paper, the authors constructed new explicit formulas for the Drazin inverse of the singular square block matrix under some conditions, where P and Q are singular square matrices over skew fields.
Abstract: In this article, we construct new explicit formulas for the Drazin inverse of $$P+Q$$ under some conditions, where P and Q are singular square matrices over skew fields. These formulas generalize some recent results in literature. By using these formulas, we establish new results for the Drazin inverse of $$2\times 2$$ block matrix under some assumptions over skew fields. Finally, illustrative numerical examples over skew fields are presented to demonstrate results of this paper.

Journal ArticleDOI
TL;DR: In this paper, the rings over which each square matrix is the sum of an idempotent matrix was studied, and it was shown that if a square matrix over a finite field is a finite field not isomorphic to any other finite field, then it is a positive ring.
Abstract: We study the rings over which each square matrix is the sum of an idempotent matrix and a $ q $ -potent matrix We also show that if $ F $ is a finite field not isomorphic to $ 𝔽_{3} $ and $ q>1 $ is odd then each square matrix over $ F $ is the sum of an idempotent matrix and a $ q $ -potent matrix if and only if $ q-1 $ is divisible by $ |F|-1 $

Posted Content
TL;DR: In this paper, a family of polynomial tetrahedron maps on the space of square matrices of arbitrary size, using a matrix refactorisation equation, which does not coincide with the standard local Yang-Baxter equation, is presented.
Abstract: We present several algebraic and differential-geometric constructions of tetrahedron maps, which are set-theoretical solutions to the Zamolodchikov tetrahedron equation. In particular, we obtain a family of new (nonlinear) polynomial tetrahedron maps on the space of square matrices of arbitrary size, using a matrix refactorisation equation, which does not coincide with the standard local Yang--Baxter equation. Liouville integrability is established for some of these maps. Also, we show how to derive linear tetrahedron maps as linear approximations of nonlinear ones, using Lax representations and the differentials of nonlinear tetrahedron maps on manifolds. We apply this construction to two nonlinear maps: a tetrahedron map obtained in [arXiv:1708.05694] in a study of soliton solutions of vector KP equations and a tetrahedron map obtained in [arXiv:2005.13574] in a study of a matrix trifactorisation problem related to a Darboux matrix associated with a Lax operator for the NLS equation. We derive parametric families of new linear tetrahedron maps, which are linear approximations for these nonlinear ones. Another result is a (nonlinear) matrix generalisation of a tetrahedron map from Sergeev's classification [arXiv:solv-int/9709006]. This matrix generalisation can be regarded as a tetrahedron map in noncommutative variables. Furthermore, we present several tetrahedron maps on arbitrary groups.

Journal ArticleDOI
01 Jan 2021
TL;DR: In this article, the weak group inverse of a square matrix is characterized based on its range space and null space, and several different characterizations of the weak-group inverse are presented by projection and the Bott-Duffin inverse.
Abstract: This paper is devoted to consider some new characteristics and properties of the weak group inverse and the weak group matrix. First, we characterize the weak group inverse of a square matrix based on its range space and null space. Also several different characterizations of the weak group inverse are presented by projection and the Bott-Duffin inverse. Then by using the core-EP decomposition, we investigate the relationships between weak group inverse and other generalized inverses. And some new characterizations of weak group matrix are obtained.

Posted Content
TL;DR: In this article, an information-theoretic method for projecting an arbitrary square matrix to the non-convex set of asymptotically stable matrices is proposed.
Abstract: We propose a principled method for projecting an arbitrary square matrix to the non-convex set of asymptotically stable matrices. Leveraging ideas from large deviations theory, we show that this projection is optimal in an information-theoretic sense and that it simply amounts to shifting the initial matrix by an optimal linear quadratic feedback gain, which can be computed exactly and highly efficiently by solving a standard linear quadratic regulator problem. The proposed approach allows us to learn the system matrix of a stable linear dynamical system from a single trajectory of correlated state observations. The resulting estimator is guaranteed to be stable and offers explicit statistical bounds on the estimation error.

Journal ArticleDOI
TL;DR: A novel pipelined algorithm for transposing non-square matrices and the corresponding architecture for this algorithm is presented, composed of a series of identical cascaded basic circuits and can be controlled via a simple control strategy based on several counters.
Abstract: In this brief, we present a novel pipelined algorithm for transposing non-square matrices and describe the corresponding architecture for this algorithm. In particular, the architecture is composed of a series of identical cascaded basic circuits and can be controlled via a simple control strategy based on several counters. The architecture is optimal in terms of both memory and latency and it achieves the theoretical minimums. Moreover, the proposed algorithm and architecture could be easily extended to $N$ -parallel implementations for matrix transposition. This architecture supports matrices whose rows and columns are integer multiples; it is mainly used for radix- $2^{s} $ butterfly algorithms using matrix transpositions. Experimental results indicate that the proposed single-path architecture can reduce the computation cycles and circuit area by a factor of 9.18% and 5.87%, respectively, for a $32\times 16$ matrix transposition computation, compared with those of a recently proposed state-of-the-art architecture for matrix transposition.

Journal ArticleDOI
TL;DR: In this paper, the class of square matrices of order n having a negative determinant and all their minors up to order n − 1 nonnegative is considered and a characterization of these matrices is presented which provides an easy test based on the Cauchon algorithm for their recognition.

Journal ArticleDOI
TL;DR: In this paper, the authors introduce a new class of matrices called Karamardian matrices, which have only zero solutions as a solution for the linear complementarity problem.
Abstract: A real square matrix $A$ is called a $Q$-matrix if the linear complementarity problem LCP$(A,q)$ has a solution for all $q \in \mathbb{R}^n$. This means that for every vector $q$ there exists a vector $x$ such that $x \geq 0, y=Ax+q\geq 0$, and $x^Ty=0$. A well-known result of Karamardian states that if the problems LCP$(A,0)$ and LCP$(A,d)$ for some $d\in \mathbb{R}^n, d >0$ have only the zero solution, then $A$ is a $Q$-matrix. Upon relaxing the requirement on the vectors $d$ and $y$ so that the vector $y$ belongs to the translation of the nonnegative orthant by the null space of $A^T$, $d$ belongs to its interior, and imposing the additional condition on the solution vector $x$ to be in the intersection of the range space of $A$ with the nonnegative orthant, in the two problems as above, the authors introduce a new class of matrices called Karamardian matrices, wherein these two modified problems have only zero as a solution. In this article, a systematic treatment of these matrices is undertaken. Among other things, it is shown how Karamardian matrices have properties that are analogous to those of $Q$-matrices. A subclass of a recently introduced notion of $P_{\#}$-matrices is shown to possess the Karamardian property, and for this reason we undertake a thorough study of $P_{\#}$-matrices and make some fundamental contributions.

Journal ArticleDOI
TL;DR: The basic properties of the numericalrange of a matrix, such as compactness and convexity, are proved to hold for the numerical range of an even-order tensor and normal tensors based on the contraction product are introduced.
Abstract: In this paper, the numerical range of an even-order tensor is defined using the norm of its square matrix unfolding. The basic properties of the numerical range of a matrix, such as compactness and convexity, are proved to hold for the numerical range of an even-order tensor. Also, we introduce normal tensors based on the contraction product. According to the Tucker decomposition, we get the numerical range of a normal tensor. Next, we introduce the singular-value decomposition (SVD) of an even-order tensor. Using this decomposition, we obtain the numerical range of such a tensor.

DOI
01 Aug 2021
TL;DR: In this article, a convergence map is obtained from solving nonlinear dynamical equation by iteration of perturbation method and study the convergence of the convergence map provided information about the stability border of dynamical aperture.
Abstract: We apply square matrix method to obtain in high speed a "convergence map", which is similar but different from frequency map [1]. The convergence map is obtained from solving nonlinear dynamical equation by iteration of perturbation method and study the convergence. The map provides information about the stability border of dynamical aperture.We compare the map with frequency map from tracking. The result indicates the new method may be applied in nonlinear lattice optimization, taking the advantage of the high speed (about 10~50 times faster) to explore x, y and the off-momentum phase space. INTRODUCTION: SQUARE MATRIX EQUATION FOR NONLINEAR DYNAMICS We consider the equations of motion of a nonlinear dynamic system with periodic structure such as Hills equation, it can be expressed by a square matrix. If we use the complex Courant-Snyder variable z = x − ip, its conjugate and powers z, z∗, z2, ... as a column Z, the one turn map can be represented by a large square matrix M using Z = MZ0. All square matrices can be transformed into Jordan form [2, 3]. A detailed description is given, e.g., in [4, 5]. For any given square matrix M, for an eigenvalue μ which is the linear tune, there is a transformation matrix U and a Jordan matrix τ so that every row of the matrix U is a (generalized) left eigenvector of M satisfying