Properties Of Such Matrices Research Articles

Theoretical and computational properties of semi-infinite quasi-Toeplitz M-matrices

A quasi-Toeplitz matrix is a semi-infinite matrix of the form A=T(a)+E, where T(a) is a Toeplitz matrix with entries (T(a))i,j=aj−i, for aj−i∈C, i,j≥1 and E is a compact correction. Quasi-Toeplitz M-matrices are encountered in the study of quadratic matrix equations arising in the analysis of a 2-dimensional Quasi-Birth-Death (QBD) stochastic process. We investigate properties of such matrices and provide conditions under which a quasi-Toeplitz matrix is an M-matrix. We show that under a mild and easy-to-check condition, an invertible quasi-Toeplitz M-matrix has a unique square root that is an M-matrix possessing quasi-Toeplitz structure. The quasi-Toeplitz structure of the square root of M-matrices provides inspirations for proving spectral properties of the quadratic matrix polynomial L(λ)=λ2A1+λA0+A−1 having quasi-Toeplitz coefficients where A1,A−1≥0 and −A0 in addition is an M-matrix. Some issues concerning the computation of square root of quasi-Toeplitz M-matrices are discussed and numerical experiments are performed.

Construction of higher-level MDS matrices in nested SPNs

This paper presents a new construction of large-scale maximum distance separable (MDS) matrices which can be used in the design of block ciphers. We extend Cauchy matrices defined over a finite field and consider these matrices on a finite commutative ring. The resulting generalized Cauchy (GC) matrices retain the advantage of a maximum branch number. We investigate the properties of such matrices in this study. First, we provide a construction of GC matrices. The proposed construction guarantees that most of the (binary) entries in a GC matrix are zero, thereby reducing the search space for cheap GC matrices considerably. Second, we minimize the number of different sub-matrices in a GC matrix to make it more suitable for lookup table implementation, we call this a compact GC matrix. We then elaborate the structure of such compact GC matrices to consider all possible matrices and thus obtain the number of compact GC matrices. Finally, we consider the involutory potential of compact GC matrices. In particular, we prove that any compact GC matrix can be modified into an involutory one through a series of column-exchanging operations and a constant matrix multiplication. Through our experiment, we discover large-scale GC matrices, which can facilitate the future design of involutory component-based ciphers. Our best constructions can reduce the number of XOR gates in the higher-level diffusion layers in Hierocrypt-L1 and Hierocrypt-3 by 35.1% and 50%, respectively, while providing the same number of active S-boxes. Moreover, compared with the existing ones, the involutory nature of the new matrices allows for equally efficient diffusion in both encryption and decryption.

Estimation of Monge matrices

Monge matrices and their permuted versions known as pre-Monge matrices naturally appear in many domains across science and engineering. While the rich structural properties of such matrices have long been leveraged for algorithmic purposes, little is known about their impact on statistical estimation. In this work, we propose to view this structure as a shape constraint and study the problem of estimating a Monge matrix subject to additive random noise. More specifically, we establish the minimax rates of estimation of Monge and pre-Monge matrices. In the case of pre-Monge matrices, the minimax-optimal least-squares estimator is not efficiently computable, and we propose two efficient estimators and establish their rates of convergence. Our theoretical findings are supported by numerical experiments.

Comparative analysis of the influence of urea additives on the properties of poly(vinyl alcohol) cryogels formed from the polymer solutions in water or dimethylsulfoxide media

Various poly(vinyl alcohol)-based gels are known for a long time and they are related to the gel materials of significant scientific and applied interests. Among different poly(vinyl alcohol) (PVA) hydrogels those formed via the so-called cryotropic gelation approach, i.e. PVA cryogels, attract special attention owing to, firstly, simplicity of their preparation and, secondly, very good properties of such matrices. The aim of this work was to compare the physicomechanical properties of PVA cryogels formed from the urea-containing aqueous and dimethylsolfoxide solutions of the polymer. It was shown that not only the solution’s nature influences greatly on the properties of the resultant cryogels, but also the low-molecule additives in the initial polymer solutions lead to the opposite effect in cryogels characteristics. The results of our investigation can expend the area of PVA cryogels usage in many fields.

On a nonlocal boundary value problem with an oblique derivative

The work studies the solvability of a nonlocal boundary value problem for the Laplace equation. The nonlocal condition is introduced using transformations in the Rn space carried out by some orthogonal matrices. Examples and properties of such matrices are given. To study the main problem, an auxiliary nonlocal Dirichlet-type problem for the Laplace equation is first solved. This problem is reduced to a vector equation whose elements are the solutions of the classical Dirichlet probem. Under certain conditions for the boundary condition coefficients, theorems on uniqueness and existence of a solution to a problem of Dirichlet type are proved. For this solution an integral representation is also obtained, which is a generalization of the classical Poisson integral. Further, the main problem is reduced to solving a non-local Dirichlet-type problem. Theorems on existence and uniqueness of a solution to the problem under consideration are proved. Using well-known statements about solutions of a boundary value problem with an oblique derivative for the classical Laplace equation, exact orders of smoothness of a problem's solution are found. Examples are also given of the cases where the theorem conditions are not fulfilled. In these cases the solution is not unique.

What the Volume of Matrices Says?

We know matrices and their transposes and we also know flip matrices. In my previous paper Matrices-One Review, I introduced transprocal matrix. Flip matrices are transpose of transprocal matrices. Now I would like to introduce water image of four matrices said above and properties of such matrices. Also we know, determinant of sum of matrices is not equal to sum of determinant of matrices. Why can’t we get equal value on addition side and additive side of determinant of matrix addition and subtraction? This question triggered me to find the reason. The basic algebra of mensuration gave ideas to retreat determinant of matrix addition and subtraction. I extent that ideas for matrices sum. Further, in 1812, French mathematician Jacques Philippe Marie Binet described how to multiply matrices. Matrices are defined on addition, subtraction and multiplication but not in division. By the inspiration of Binet, I would like to describe how to do divisions on matrices. This idea is derived from division of fractions. In division of fraction, reciprocal of divisor fraction multiplies with dividend fraction. I do the same in division on matrices with some modifications. By this way, we could find quotient matrix and remainder matrix which satisfy division algorithm. So we could say, determinant of division of dividend matrix and divisor matrix is equal to division of determinant of dividend matrix and determinant of divisor matrix.

Predicative logical matrices

In classical linear algebra the machine of matrices is widely used. But the classic linear algebra deals with continuous objects. Logical algebra, built by analogy with the classical linear algebra, builds the same models using discrete objects that have logical structure and obey the relevant laws. This leads to a significant difference in the functioning of the constructed models. This article is devoted to matrices, as elements for which the elementary logical elements are taken, namely the finite predicates of any quality of variables. In the work investigated the properties of such matrices and features of their application. Basic operations on such matrices are also considered. Besides the usual operations that take place in classical linear algebra, logical structures allow to perform this several operations.

On the inverse eigenvalue problem for periodic Jacobi matrices

ABSTRACT The problem of reconstructing a matrix with a specific structure from a partial or total spectral data is known as inverse eigenvalue problem which arises in a variety of applications. In this paper, we study a partially described inverse eigenvalue problem of periodic Jacobi matrices and prove some spectral properties of such matrices. The problem involves the reconstruction of the matrix by one eigenvalue of each of its leading principal submatrices and one eigenvector of the required matrix and one more additional piece of information. The conditions for solvability of the problem are presented and finally an algorithm and some numerical results are given.

Matrices whose inverses are tridiagonal, band or block-tridiagonal and their relationship with the covariance matrices of a random Markov process

The article discusses the matrices of the form A1n, Amn, AmN, whose inverses are: tridiagonal matrix A-1n (n - dimension of the A-mn matrix), banded matrix A-mn (m is the half-width band of the matrix) or block-tridiagonal matrix A-m N (N = n x m - full dimension of the block matrix; m - the dimension of the blocks) and their relationships with the covariance matrices of measurements with ordinary (simple) Markov Random Processes (MRP), multiconnected MRP and vector MRP, respectively. Such covariance matrices frequently occur in the problems of optimal filtering, extrapolation and interpolation of MRP and Markov Random Fields (MRF). It is shown, that the structures of the matrices A1n, Amn, AmN have the same form, but the matrix elements in the first case are scalar quantities; in the second case matrix elements represent a product of vectors of dimension m; and in the third case, the off-diagonal elements are the product of matrices and vectors of dimension m. The properties of such matrices were investigated and a simple formulas of their inverses were found. Also computational efficiency in the storage and the inverse of such matrices have been considered. To illustrate the acquired results, an example on the covariance matrix inversions of two-dimensional MRP is given.

Design and evaluation of mucoadhesive oral films containing sodium hyaluronate using multivariate data analysis

Context: Mucoadhesive oral films, with their prolonged residence time at the site of application, offer a promising approach for protection of the oral lesion surface. The addition of sodium hyaluronate of different molecular weights as a second mucoadhesive polymer into the film matrix could positively influence the physico-mechanical and mucoadhesive properties of films.Objective: The aim of this study was to investigate the formulation of a monolayered film matrix containing varying amounts of sodium hyaluronate and to test the properties of such matrices by applying different characterization methods.Materials and methods: Film matrix was composed of two mucoadhesive polymers, carmellose sodium and sodium hyaluronate, plasticized with glycerol. Resulting films were characterized with regard to their viscosity and physico-mechanical properties.Results and discussion: Multivariate data analysis was employed to evaluate the influence of varying amounts of mucoadhesive polymers on the main mucoadhesive oral films’ properties. The lower content of sodium hyaluronate caused improvements in mechanical properties and residence time on the artificial oral mucosa, both of which are the main characteristics that determine the quality of the final product.Conclusions: The best results were obtained by samples containing carmellose sodium with a small amount of sodium hyaluronate (about 0.5% in casting dispersion).

Theoretical basis of the iterative process of the joint assessment of difficulties in tasks and levels of training students

In this paper, we study the iterative process of the joint numerical assessment of levels of training students and difficulties in tasks of diagnostic tools using the dichotomous response matrix A of size N × M with allowance for the contribution of tasks of different difficulty to the assessments obtained. It is shown that not for any matrix A there exist infinite iterative sequences, and in the case of their existence, they do not always converge. A wide range of sufficient conditions for their convergence have been obtained, which are based on the following: (1) the matrix A contains at least three different columns; (2) if one places the columns of the matrix A in non-decreasing order of column sums, then for any position of the vertical dividing line between the columns there exists a row, which has at least one unity to the left of the line and at least one zero to the right of the line. It is established that the response matrix A obtained as a result of testing reliability satisfies these two conditions. The properties of such matrices have been studied. In particular, the equivalence of the above-mentioned conditions of primitiveness of the square matrix B of order M with the entries \({b_{ij}} = \sum\limits_{\ell = 1}^N {(1 - {a_{\ell i}}){a_{\ell i}}} \) has been proved. Using the matrix analysis, we have proved that the primitiveness of the matrix B ensures the convergence of iterative sequences, as well as independence of their limits of the choice of the initial approximation. We have estimated the rate of convergence of these sequences and found their limits.

BIOLOGICALLY INSPIRED FILTERS UTILIZING SPECTRAL PROPERTIES OF TOEPLITZ-BLOCK-TOEPLITZ MATRICES

The construction of filters arising from linear neural networks with feed-backward excitatory-inhibitory connections is presented. Spatially invariant coupling between neurons and the distribution of neuron-receptor units in the form of a uniform square grid yield the TBT (Toeplitz-Block-Toeplitz) connection matrix. Utilizing the relationship between spectral properties of such matrices and their generating functions, the method for construction of recurrent linear networks is addressed. By appropriately bounding the generating function, the connection matrix eigenvalues are kept in the desired range allowing for large matrix inverse to be approximated by a convergent power series. Instead of matrix inversion, the single pass convolution with the filter obtained from the network connection weights is applied when solving the network. For the case of inter-neuron coupling in the form of a function that is expandable in a Fourier series in polar angle, the network response filter is shown to be steerable.

Almost Definite Matrices Revisited

A real matrix A is called as an almost definite matrix if ⟨x, Ax⟩ = 0 ⇒ Ax = 0. This notion is revisited. Many basic properties of such matrices are established. Several characterizations for a matrix to be an almost definite matrix are presented. Comparisons of certain properties of almost definite matrices with similar properties for positive definite or positive semidefinite matrices are brought to the fore. Interconnections with matrix classes arising in the theory of linear complementarity problems are discussed briefly.

Matrices with totally positive powers and their generalizations

In this paper, eventually totally positive matrices (i.e. matrices all whose powers starting at some point are totally positive) are studied. We present a new approach to eventual total positivity which is based on the theory of eventually positive matrices. We mainly focus on the spectral properties of such matrices. We also study eventually J-sign-symmetric matrices and matrices, whose powers are P -matrices. Mathematics subject classification (2010): Primary 15A48; Secondary 15A18, 15A75.

Cryptographically significant MDS matrices based on circulant and circulant-like matrices for lightweight applications

MDS matrices incorporate diffusion layers in block ciphers and hash functions. MDS matrices are in general not sparse and have a large description and thus induce costly implementations both in hardware and software. It is also nontrivial to find MDS matrices which could be used in lightweight cryptography. In the AES MixColumn operation, a circulant MDS matrix is used which is efficient as its elements are of low hamming weights, but no general constructions and study of MDS matrices from d×d circulant matrices for arbitrary d is available in the literature. In a SAC 2004 paper, Junod et al. constructed a new class of efficient matrices whose submatrices were circulant matrices and they coined the term circulating-like matrices for these new class of matrices. We call these matrices as Type-I circulant-like matrices. In this paper we introduce a new type of circulant-like matrices which are involutory by construction and we call them Type-II circulant-like matrices. We study the MDS properties of d×d circulant, Type-I and Type-II circulant-like matrices and construct new and efficient MDS matrices which are suitable for lightweight cryptography for d up to 8. We also consider orthogonal and involutory properties of such matrices and study the construction of efficient MDS matrices whose inverses are also efficient. We explore some interesting and useful properties of circulant, Type-I and Type-II circulant-like matrices which are prevalent in many parts of mathematics and computer science.

Random Unitary Matrices Associated to a Graph

We analyze composed quantum systems consisting of $k$ subsystems, each described by states in the $n$-dimensional Hilbert space. Interaction between subsystems can be represented by a graph, with vertices corresponding to individual subsystems and edges denoting a generic interaction, modeled by random unitary matrices of order $n^2$. The global evolution operator is represented by a unitary matrix of size $N=n^k$. We investigate statistical properties of such matrices and show that they display spectral properties characteristic to Haar random unitary matrices provided the corresponding graph is connected. Thus basing on random unitary matrices of a small size $n^2$ one can construct a fair approximation of large random unitary matrices of size $n^{k}$. Graph--structured random unitary matrices investigated here allow one to define the corresponding structured ensembles of random pure states.

Large N (=3) neutrinos and random matrix theory

Abstract The large N limit has been successfully applied to QCD, leading to qualitatively correct results even for N = 3. In this work, we propose to treat the number N = 3 of Standard Model generations as a large number. Specifically, we apply this idea to the neutrino anarchy scenario and study neutrino physics using Random Matrix Theory, finding new results in both areas. For neutrino physics, we obtain predictions for the masses and mixing angles as a function of the generation number N. The Seesaw mechanism produces a hierarchy of order 1/N 3 between the lightest and heaviest neutrino, and a θ 13 mixing angle of order 1/N, in parametric agreement with experimental data when N goes to 3. For Random Matrix Theory, this motivates the introduction of a new type of ensemble of random matrices, the “Seesaw ensemble.” Basic properties of such matrices are studied, including the eigenvalue density and the interpretation as a Coulomb gas system. Besides its mathematical interest, the Seesaw ensemble may be useful in random systems where two hierarchical scales exist.

Identities induced by Riordan arrays

Historically, there exist two versions of the Riordan array concept. The older one (better known as recursive matrix) consists of bi-infinite matrices ( d n , k ) n , k ∈ Z ( k > n implies d n , k = 0 ), deals with formal Laurent series and has been mainly used to study algebraic properties of such matrices. The more recent version consists of infinite, lower triangular arrays ( d n , k ) n , k ∈ N ( k > n implies d n , k = 0 ), deals with formal power series and has been used to study combinatorial problems. Here we show that every Riordan array induces two characteristic combinatorial sums in three parameters n , k , m ∈ Z . These parameters can be specialized and generate an indefinite number of other combinatorial identities which are valid in the bi-infinite realm of recursive matrices.

Generalized Leibniz functional matrices and factorizations of some well-known matrices

In this paper, we introduce the generalized Leibniz functional matrices and study some algebraic properties of such matrices. To demonstrate applications of these properties, we derive several novel factorization forms of some well-known matrices, such as the complete symmetric polynomial matrix and the elementary symmetric polynomial matrix. In addition, by applying factorizations of the generalized Leibniz functional matrices, we redevelop the known results of factorizations of Stirling matrices of the first and second kind and the generalized Pascal matrix.

Effect of calcium silicate sources on geopolymerisation

Seven different calcium silicate materials were used to investigate the role of calcium in geopolymerisation. At low alkalinity, the compressive strength of matrices prepared with predominantly amorphous calcium silicates (blast furnace slag) or containing crystalline phases specifically manufactured for reactivity (cement) is much higher than when the calcium is supplied as crystalline silicate minerals. The compressive strength of matrices containing natural (crystalline) calcium silicates improves with increasing alkalinity, however the opposite trend is observed in matrices synthesised with processed calcium silicate sources. The difference in compressive strength between matrices synthesised using different calcium silicate sources is significantly reduced at high alkalinity. An insufficient amount of calcium is dissolved from crystalline calcium silicates at relatively low alkalinity to enable formation of calcium silicate hydrate in coexistence with the aluminosilicate geopolymeric gel, and this leads to the poor mechanical properties of such matrices. At high alkalinity, calcium plays a lesser role in affecting the nature of the final binder, as it forms precipitates rather than hydrated gels. Thus, the different calcium silicate sources will not have a major impact on the mechanical properties of these matrices. The effects of different calcium silicates on geopolymerisation are therefore seen to depend most significantly on two factors: the crystallinity of the calcium silicate source, and the alkalinity of the activating solution used.