Class Of Matrices Research Articles

Idempotent factorization on some matrices over quadratic integer rings

In 2020, Cossu and Zanardo raised a problem on the idempotent factorization of singular matrices in the form ( p z z ¯ ‖ z ‖ / p ) , where p is a prime integer which is irreducible but not prime in the ring of integers O K , with K a real quadratic number field, and z ∈ O K is such that 〈 p , z 〉 is a non-principal ideal. In this paper, we provide some classes of matrices that factor in a prescribed way and some other classes that do not. We further show that there are matrices in the above form that cannot be written as a product of two idempotent matrices.

HODLR2D: A New Class of Hierarchical Matrices

HODLR2D: A New Class of Hierarchical Matrices

The QARMAv2 Family of Tweakable Block Ciphers

We introduce the QARMAv2 family of tweakable block ciphers. It is a redesign of QARMA (from FSE 2017) to improve its security bounds and allow for longer tweaks, while keeping similar latency and area. The wider tweak input caters to both specific use cases and the design of modes of operation with higher security bounds. This is achieved through new key and tweak schedules, revised S-Box and linear layer choices, and a more comprehensive security analysis. QARMAv2 offers competitive latency and area in fully unrolled hardware implementations.Some of our results may be of independent interest. These include: new MILP models of certain classes of diffusion matrices; the comparative analysis of a full reflection cipher against an iterative half-cipher; our boomerang attack framework; and an improved approach to doubling the width of a block cipher.

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 the hyperbolicity of the Krein space numerical range

In this paper a necessary and sufficient condition for hyperbolicity of the indefinite numerical range is established. As a consequence, an indefinite version of Brown–Spitkovsky theorem stating the ellipticity of the numerical range of certain tridiagonal matrices is revisited. This result leads to necessary and sufficient conditions for hyperbolicity of indefinite numerical ranges of new classes of tridiagonal matrices.

On the Total Positivity and Accurate Computations of r-Bell Polynomial Bases

A new class of matrices defined in terms of r-Stirling numbers is introduced. These r-Stirling matrices are totally positive and determine the linear transformation between monomial and r-Bell polynomial bases. An efficient algorithm for the computation to high relative accuracy of the bidiagonal factorization of r-Stirling matrices is provided and used to achieve computations to high relative accuracy for the resolution of relevant algebraic problems with collocation, Wronskian, and Gramian matrices of r-Bell bases. The numerical experimentation confirms the accuracy of the proposed procedure.

On Some Sequence Spaces via q-Pascal Matrix and Its Geometric Properties

We develop some new sequence spaces 𝓁p(P(q)) and 𝓁∞(P(q)) by using q-Pascal matrix P(q). We discuss some topological properties of the newly defined spaces, obtain the Schauder basis for the space 𝓁p(P(q)) and determine the Alpha-(α-), Beta-(β-) and Gamma-(γ-) duals of the newly defined spaces. We characterize a certain class (𝓁p(P(q)),X) of infinite matrices, where X∈{𝓁∞,c,c0}. Furthermore, utilizing the proposed results, we characterize certain other classes of infinite matrices. We also examine some geometric properties, like the approximation property, Dunford–Pettis property, Hahn–Banach extension property, and Banach–Saks-type p property of the spaces 𝓁p(P(q)) and 𝓁∞(P(q)).

Williamson type Hadamard matrices with circulant components

It has been shown that Williamson Hadamard matrices of order 4n do not exist for n=35,47,53,59. We consider the class of Williamson type matrices with circulant (but not necessarily symmetric) components when n=p is an odd prime or n=pq is the product of distinct odd primes each of which is a generator modulo the other, showing that any such matrix may be derived from a Williamson matrix by a common cyclic shift of the components. In particular, this means that there are no Williamson type matrices with circulant components for n=35,47,53,59.

Some Properties of Hessenberg Matrices with Conditional Elements

Hessenberg matrices are a class of structured matrices that play a fundamental role in various areas of mathematics and scientific computing. These matrices have diverse applications in numerical analysis, optimization, graph theory, and quantum mechanics, among others. This paper provides the determinants of the conditional Hessenberg matrices whose elements are the generalized bi-periodic Fibonacci numbers. Moreover, we find the inverse of some special type matrices which help us to find the determinant of the conditional Hessenberg matrices.

Multislant matrices and Jacobi–Trudi determinants over finite fields

The problem of counting the Fq-valued points of a variety has been well-studied from algebro-geometric, topological, and combinatorial perspectives. We explore a combinatorially flavored version of this problem studied by Anzis et al. [1], which is similar to work of Kontsevich [7], Elkies [4], and Haglund [5].Anzis et al. considered the question: what is the probability that the determinant of a Jacobi-Trudi matrix vanishes if the variables are chosen uniformly at random from a finite field? They gave a formula for various partitions such as hooks, staircases, and rectangles. We give a formula for partitions whose parts form an arithmetic progression, verifying and generalizing one of their conjectures. More generally, we compute the probability of the determinant vanishing for a class of matrices (“multislant matrices”) made of Toeplitz blocks with certain properties.We furthermore show that the determinant of a skew Jacobi-Trudi matrix is equidistributed across the finite field if the skew partition is a ribbon.

Structure and Approximation Properties of Laplacian-Like Matrices

Many of today’s problems require techniques that involve the solution of arbitrarily large systems Atextbf{x}=textbf{b}. A popular numerical approach is the so-called Greedy Rank-One Update Algorithm, based on a particular tensor decomposition. The numerical experiments support the fact that this algorithm converges especially fast when the matrix of the linear system is Laplacian-Like. These matrices that follow the tensor structure of the Laplacian operator are formed by sums of Kronecker product of matrices following a particular pattern. Moreover, this set of matrices is not only a linear subspace it is a Lie sub-algebra of a matrix Lie Algebra. In this paper, we characterize and give the main properties of this particular class of matrices. Moreover, the above results allow us to propose an algorithm to explicitly compute the orthogonal projection onto this subspace of a given square matrix A in mathbb {R}^{Ntimes N}.

A Riemann–Hilbert Approach to the Perturbation Theory for Orthogonal Polynomials: Applications to Numerical Linear Algebra and Random Matrix Theory

Abstract We establish a new perturbation theory for orthogonal polynomials using a Riemann–Hilbert approach and consider applications in numerical linear algebra and random matrix theory. This new approach shows that the orthogonal polynomials with respect to two measures can be effectively compared using the difference of their Stieltjes transforms on a suitably chosen contour. Moreover, when two measures are close and satisfy some regularity conditions, we use the theta functions of a hyperelliptic Riemann surface to derive explicit and accurate expansion formulae for the perturbed orthogonal polynomials. In contrast to other approaches, a key strength of the methodology is that estimates can remain valid as the degree of the polynomial grows. The results are applied to analyze several numerical algorithms from linear algebra, including the Lanczos tridiagonalization procedure, the Cholesky factorization, and the conjugate gradient algorithm. As a case study, we investigate these algorithms applied to a general spiked sample covariance matrix model by considering the eigenvector empirical spectral distribution and its limits. For the first time, we give precise estimates on the output of the algorithms, applied to this wide class of random matrices, as the number of iterations diverges. In this setting, beyond the first order expansion, we also derive a new mesoscopic central limit theorem for the associated orthogonal polynomials and other quantities relevant to numerical algorithms.

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.

On adequacy of full matrices

This paper deals with the following question:whether a ring of matrices or classes of matrices over an adequate ring or elementary divisor ring inherits the property of adequacy?
 The property to being adequate in matrix rings over adequate and commutative elementary divisor rings is studied.Let us denote by $\mathfrak{A}$ and $\mathfrak{E}$ an adequate and elementary divisor domains, respectively. Also $\mathfrak{A}_2$ and $\mathfrak{E}_2$ denote a rings of $2 \times 2$ matrices over them. We prove that full nonsingular matrices from $\mathfrak{A}_2$ are adequate in $\mathfrak{A}_2$ and full singular matrices from $\mathfrak{E}_2$ are adequate in the set of full matrices in $\mathfrak{E}_2$.

Edge Behavior of Higher Complex-Dimensional Determinantal Point Processes

As recently proved in generality by Hedenmalm and Wennman, it is a universal behavior of complex random normal matrix models that one finds a complementary error function behavior at the boundary (also called edge) of the droplet as the matrix size increases. Such behavior is seen both in the density of the eigenvalues and the correlation kernel, where the Faddeeva plasma kernel emerges. These results are neatly expressed with the help of the outward unit normal vector on the edge. We prove that such universal behaviors transcend this class of random normal matrices, being also valid in a specific “elliptic” class of determinantal point processes defined on mathbb C^d, which are higher dimensional generalizations of the determinantal point processes describing the eigenvalues of the complex Ginibre ensemble and the complex elliptic Ginibre ensemble. These models describe a system of particles in mathbb C^d with mutual repulsion, that are confined to the origin by an external field {mathscr {V}}(z) = |z|^2 - tau {text {Re}}(z_1^2+ldots +z_d^2), where 0le tau <1. Their average density of particles converges to a uniform law on a 2d-dimensional ellipsoidal region. It is on the hyperellipsoid bounding this region that we find a complementary error function behavior and the Faddeeva plasma kernel. To the best of our knowledge, this is the first instance of the Faddeeva plasma kernel emerging in a higher dimensional model. The results provide evidence for a possible edge universality theorem for determinantal point processes on mathbb C^d.

Distributed ℋ 2 -Matrices for Boundary Element Methods

Standard discretization techniques for boundary integral equations, e.g., the Galerkin boundary element method, lead to large densely populated matrices that require fast and efficient compression techniques like the fast multipole method or hierarchical matrices. If the underlying mesh is very large, running the corresponding algorithms on a distributed computer is attractive, e.g., since distributed computers frequently are cost-effective and offer a high accumulated memory bandwidth. Compared to the closely related particle methods, for which distributed algorithms are well-established, the Galerkin discretization poses a challenge, since the supports of the basis functions influence the block structure of the matrix and therefore the flow of data in the corresponding algorithms. This article introduces distributed ℋ 2 -matrices, a class of hierarchical matrices that is closely related to fast multipole methods and particularly well-suited for distributed computing. While earlier efforts required the global tree structure of the ℋ 2 -matrix to be stored in every node of the distributed system, the new approach needs only local multilevel information that can be obtained via a simple distributed algorithm, allowing us to scale to significantly larger systems. Experiments show that this approach can handle very large meshes with more than 130 million triangles efficiently.

Central limit theorem for linear spectral statistics of block-Wigner-type matrices

Motivated by the stochastic block model, we investigate a class of Wigner-type matrices with certain block structures and establish a CLT for the corresponding linear spectral statistics (LSS) via the large-deviation bounds from local law and the cumulant expansion formula. We apply the results to the stochastic block model. Specifically, a class of renormalized adjacency matrices will be block-Wigner-type matrices. Further, we show that for certain estimator of such renormalized adjacency matrices, which will be no longer Wigner-type but share long-range non-decaying weak correlations among the entries, the LSS of such estimators will still share the same limiting behavior as those of the block-Wigner-type matrices, thus enabling hypothesis testing about stochastic block model.

A divide-and-conquer method for constructing a pseudo-Jacobi matrix from mixed given data

For the given signature operator H=Ir⊕−In−r, a pseudo-Jacobi matrix is a self-adjoint matrix relatively to a symmetric bilinear form 〈⋅,⋅〉H, and it is the counterpart of a classical Jacobi matrix to the indefinite scalar product space setting. In this article, we consider an inverse eigenvalue problem for this class of matrices. The main concern is to construct an n×n pseudo-Jacobi matrix from a prescribed n-tuple of distinct real numbers and a Jacobi matrix of order not less than ⌊n2⌋, such that its spectrum is this tuple and the given Jacobi matrix is its trailing principal submatrix. A divide-and-conquer scheme is used to solve this problem, and a necessary and sufficient condition under which the problem is solvable is presented. A numerical algorithm is designed to solve this pseudo-Jacobi matrix inverse eigenvalue problem according to the obtained results. Some illustrative numerical examples are also given to test the reconstructive algorithm.

Pseudo-Commutation Classes of Complex Matrices and Their Decomplexification

Pseudo-Commutation Classes of Complex Matrices and Their Decomplexification

On the class of matrices with rows that weakly decrease cyclicly from the diagonal

We consider n×n real-valued matrices A=(aij) satisfying aii≥ai,i+1≥…≥ain≥ai1≥…≥ai,i−1 for i=1,…,n. With such a matrix A we associate a directed graph G(A). We prove that the solutions to the system A⊤x=λe, with λ∈R and e the vector of all ones, are linear combinations of ‘fundamental’ solutions to A⊤x=e and vectors in ker⁡A⊤, each of which is associated with a closed strongly connected component (SCC) of G(A). This allows us to characterize the sign of det⁡A in terms of the number of closed SCCs and the solutions to A⊤x=e. In addition, we provide conditions for A to be a P-matrix.