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In these notes we generalize the notion of a (pseudo) metric measuring the distance of two points, to a (pseudo) n-metric which assigns a value to a tuple of n ≥ 2 points. Some elementary properties of pseudo n-metrics are provided and their construction via exterior products is investigated. We discuss some examples from the geometry of Euclidean vector spaces leading to pseudo n-metrics on the unit sphere, on the Stiefel manifold, and on the Grassmann manifold. Further, we construct a pseudo n-metric on hypergraphs and discuss the problem of generalizing the Hausdorff metric for closed sets to a pseudo n-metric. [ABSTRACT FROM AUTHOR]

Let $ p_1 \lt p_2 \lt \cdots \lt p_n $ p 1 < p 2 < ⋯ < p n be positive numbers and r a non-negative real number. The Loewner matrix associated with the function $ x^{r+1} $ x r + 1 given by $ L_{r+1}=\begin {bmatrix}\frac {p_i^{r+1}-p_j^{r+1}}{p_i-p_j}\end {bmatrix} $ L r + 1 = [ p i r + 1 − p j r + 1 p i − p j ] and matrix $ P_r=[\begin {smallmatrix}{(p_i+p_j)^r}\end {smallmatrix}] $ P r = [ (p i + p j) r ] (the Hadamard inverse of rth Hadamard power of well-known Cauchy matrix) have same inertia. A question was left open in Inertia of Loewner matrices. Indiana Univ Math J. 2016;65(4):1251–1261 by Bhatia, Friedland and Jain to find a connection between these two matrix families. We aim to answer this question firmly in terms of a congruence relation between $ L_{r+1} $ L r + 1 and $ P_r $ P r . Indeed, a non-singular matrix X over $ \mathbb {R} $ R is explicitly obtained such that $ X'P_rX=L_{r+1} $ X ′ P r X = L r + 1 in this paper. [ABSTRACT FROM AUTHOR]

The stable index of a 0–1 matrix A is defined to be the smallest integer k such that $ A^{k+1} $ A k + 1 is not a 0–1 matrix if such an integer exists; otherwise the stable index of A is defined to be infinity. We characterize the set of stable indices of 0–1 matrices with a given order. [ABSTRACT FROM AUTHOR]

Let p be an odd prime and let a , b ∈ Z with p ∤ a b . In this paper,we mainly evaluate T p (δ) (a , b , x) : = det x + tan π a j 2 + b k 2 p δ ⩽ j , k ⩽ (p - 1) / 2 (δ = 0 , 1). For example, in the case p ≡ 3 (mod 4) , we show that T p (1) (a , b , 0) = 0 and T p (0) (a , b , x) = 2 (p - 1) / 2 p (p + 1) / 4 if (ab p) = 1 , p (p + 1) / 4 if (ab p) = - 1 , where (· p) is the Legendre symbol. When (- a b p) = - 1 , we also evaluate the determinant det [ x + cot π a j 2 + b k 2 p ] 1 ⩽ j , k ⩽ (p - 1) / 2. In addition, we pose several conjectures one of which states that for any prime p ≡ 3 (mod 4) , there is an integer x p ≡ 1 (mod p) such that det sec 2 π (j - k) 2 p 0 ⩽ j , k ⩽ p - 1 = - p (p + 3) / 2 x p 2. [ABSTRACT FROM AUTHOR]

This paper wants to increase our understanding and computational know-how for time-varying matrix problems and Zhang Neural Networks. These neural networks were invented for time or single parameter-varying matrix problems around 2001 in China and almost all of their advances have been made in and most still come from its birthplace. Zhang Neural Network methods have become a backbone for solving discretized sensor driven time-varying matrix problems in real-time, in theory and in on-chip applications for robots, in control theory and other engineering applications in China. They have become the method of choice for many time-varying matrix problems that benefit from or require efficient, accurate and predictive real-time computations. A typical discretized Zhang Neural Network algorithm needs seven distinct steps in its initial set-up. The construction of discretized Zhang Neural Network algorithms starts from a model equation with its associated error equation and the stipulation that the error function decrease exponentially fast. The error function differential equation is then mated with a convergent look-ahead finite difference formula to create a distinctly new multi-step style solver that predicts the future state of the system reliably from current and earlier state and solution data. Matlab codes of discretized Zhang Neural Network algorithms for time varying matrix problems typically consist of one linear equations solve and one recursion of already available data per time step. This makes discretized Zhang Neural network based algorithms highly competitive with ordinary differential equation initial value analytic continuation methods for function given data that are designed to work adaptively. Discretized Zhang Neural Network methods have different characteristics and applicabilities than multi-step ordinary differential equations (ODEs) initial value solvers. These new time-varying matrix methods can solve matrix-given problems from sensor data with constant sampling gaps or from functional equations. To illustrate the adaptability of discretized Zhang Neural Networks and further the understanding of this method, this paper details the seven step set-up process for Zhang Neural Networks and twelve separate time-varying matrix models. It supplies new codes for seven of these. Open problems are mentioned as well as detailed references to recent work on discretized Zhang Neural Networks and time-varying matrix computations. Comparisons are given to standard non-predictive multi-step methods that use initial value problems ODE solvers and analytic continuation methods. [ABSTRACT FROM AUTHOR]

Let p 1 , p 2 , ... , p n , (n ≥ 2) , be distinct positive numbers and r > 0 . We propose to study a comparison of the positivity properties of two families of matrices, K r + 1 = p i r + 1 + p j r + 1 p i + p j and B r = | p i - p j | r in full. Indeed, Bhatia and Jain (Spectr. Theory 5(1):71–87, 2015) studied about B r and carried out rigorous analysis on the study of K r . They conjectured therein that inertia of K r + 1 and B r are same for all r > 0 . We settle a congruence relation between these two families in this paper. [ABSTRACT FROM AUTHOR]

Recently, solving tensor equations (or multilinear systems) has attracted a lot of attention. This paper investigates the tensor form of the Bi-CGSTAB and Bi-CRSTAB methods, by employing Kronecker product, vectorization, and bilinear operator, to solve the generalized coupled Sylvester tensor equations ∑ i = 1 n (X × 1 A i 1 × 2 A i 2 + Y × 1 B i 1 × 2 B i 2) = E 1 , ∑ i = 1 n (X × 1 C i 1 × 2 C i 2 + Y × 1 D i 1 × 2 D i 2) = E 2 , with no matricization. Also some properties of the new methods are presented. By applying multilinear operator, the proposed methods are extended to the general form. Some numerical examples are provided to compare the efficiency of the investigated methods with some existing popular algorithms. Finally, some concluding remarks are given. [ABSTRACT FROM AUTHOR]

We formulate the Euclidean algorithm using directed graphs on integer points in the plane and operations on a particular semigroup of two by two matrices. The properties of the graphs and the semigroup provide surprisingly effective tools for solving certain classical diophantine equations, among other applications. [ABSTRACT FROM AUTHOR]

In this work we investigate when an n × n Jordan matrix can be written as the sum of two strictly n-zero matrices. In particular, we show that if F is an algebraically closed field of characteristic zero and A is an n × n matrix over F with tr (A) = 0 , then A is the sum of two strictly n-zero matrices. [ABSTRACT FROM AUTHOR]

Let C(A)=A∘A−T{\mathcal{C}}\left(A)=A\circ {A}^{-T} be the combined matrix of an invertible matrix AA, where ∘\circ means the Hadamard product of matrices. In this work, we study the combined matrix of a nonsingular matrix, which is an HH-matrix whose comparison matrix is singular. In particular, we focus on C(A){\mathcal{C}}\left(A) when AA is diagonally equipotent, and we study whether C(A){\mathcal{C}}\left(A) is an HH-matrix and to which class it belongs. Moreover, we give some properties on the diagonal dominance of these matrices and on their comparison matrices.

In this paper, we study certain determinants involving tangent functions. We prove that for any odd number n ⩾ 3 and s ∈ Z + , the determinant D (n , s) = det (tan s ⁡ π j + k n) 1 ⩽ j , k ⩽ n − 1 ∈ n n − 2 Z . In the special case s = 2, we confirm a conjecture raised by Zhi-Wei Sun in 2019. [ABSTRACT FROM AUTHOR]

A ring R is (strongly) weakly nil clean if every element in R is the sum or difference of a nilpotent and an idempotent (that commutes). In this note, we show that if R is strongly nil clean such that J(R) is locally nilpotent, then M n (R) is weakly nil clean. We also give a characterization of strongly weakly nil cleanness of the group ring RG where R is a ring and G is a group, and a characterization of weakly nil cleanness of the group ring RG, when R is a ring and G is a nilpotent group. If A and B are two strongly weakly nil-clean k-algebras (k is a commutative ring) such that J(A) and J(B) are locally nilpotent, then A ⊗ k B is a strongly weakly nil-clean k-algebra. This gives an answer to the question posed in [21]. [ABSTRACT FROM AUTHOR]

This article aims to interpret some Lie triple derivations of Tensor algebras T ⊗ F (a , b F) , using generalized quaternion algebra (a , b F) over a field F and assuming T as an F-linear associative algebra. The study concludes with complete description of all Lie triple derivations of A ⊗ FL. It is pertinent to mention here that A is assumed to be a finite-dimensional commutative algebra and L is a simple Lie algebra over an algebraically closed field. [ABSTRACT FROM AUTHOR]

Inspired by circular complex interval arithmetic, an arithmetic for closed balls in ℝn is pursued. In this sense, the properties of certain operations on closed balls in ℝn, some of which related either to the Hadamard product of vectors or to the 2-fold vector cross product when n ∈ {3,7}, are studied. In particular, known results for operations on closed balls in ℂ, which can be identified with ℝ2, are extended to closed balls in ℝn. [ABSTRACT FROM AUTHOR]

We give effective lower and upper bounds on the maximal generating degree d (I n ¯) and the Castelnuovo–Mumford regularity reg (I n ¯) of the integral closure of a power I n of a monomial ideal I for all n and determine the leading coefficient of the asymptotically linear functions d (I n ¯) and reg (I n ¯) . A number n 0 is also given such that d (I n ¯) becomes a linear function of n when n ≥ n 0 . [ABSTRACT FROM AUTHOR]

Tucker decomposition is a popular technique for many data analysis and machine learning applications. Finding a Tucker decomposition is a nonconvex optimization problem. As the scale of the problems increases, local search algorithms such as stochastic gradient descent have become popular in practice. In this paper, we characterize the optimization landscape of the Tucker decomposition problem. In particular, we show that if the tensor has an exact Tucker decomposition, for a standard nonconvex objective of Tucker decomposition, all local minima are also globally optimal. We also give a local search algorithm that can find an approximate local (and global) optimal solution in polynomial time. [ABSTRACT FROM AUTHOR]

For a nonnegative integer n, let P n be the vector space of polynomials of degree at most n, equipped with the inner product ⟨ f (x) , g (x) ⟩ = ∫ - 1 1 f (x) g (x) d x . Let B = { x n , x n - 1 , ... , 1 } , an ordered basis of P . Let { P ← n n (x) , P ← n - 1 n (x) , ⋯ , P ← 0 n (x) } be the orthogonal basis of P n obtained by applying the Gram–Schmidt procedure to B , with the resulting polynomials normalized by the condition that P ← k n (1) = 1 . We call these polynomials reverse Legendre polynomials. In this paper, we give explicit formulas for the reverse Legendre polynomials { P ← k n (x) } and determine some of their properties. [ABSTRACT FROM AUTHOR]

Lyapunov stability theory for smooth nonlinear autonomous dynamical systems is presented in terms of Geometric Algebra. The system is described by a smooth nonlinear state vector differential equation, driven by a vector field in R n . The level sets of the scalar Lyapunov function candidate are assumed to be compact smooth vector manifolds in R n . The level sets induce an associated global foliation of the state space. On any leaf of this foliation, a geometric subalgebra is naturally attached to the corresponding tangent vector space of the smooth vector manifold. The pseudoscalar (field) of this subalgebra completely characterizes the tangent space. Asymptotic stability of the system equilibria is described in terms of equilibria of, easily computable, rejection vector fields with respect to the pseudoscalar field. Nonexistence of invariant sets of the Lyapunov function directional derivative, along the defining vector field, are also tested using a simple tangency condition. Several illustrative examples are presented. [ABSTRACT FROM AUTHOR]

Let B (H) be the algebra of all bounded linear operators acting on a complex Hilbert space H . In this paper, we investigate the general form of bijective unital maps Ψ : B (H) → B (H) , not necessarily linear, that preserve Drazin invertible 2 × 2 operator matrices of index one with generalized Schur complements which are either invertible or Drazin invertible of index one. [ABSTRACT FROM AUTHOR]

We show that the matrix query language MATLANG corresponds to a natural fragment of the positive relational algebra on K-relations. The fragment is defined by introducing a composition operator and restricting K-relation arities to 2. We then proceed to show that MATLANG can express all matrix queries expressible in the positive relational algebra on K-relations, when intermediate arities are restricted to 3. Thus we offer an analogue, in a model with numerical data, to the situation in classical logic, where the algebra of binary relations is equivalent to first-order logic with three variables. [ABSTRACT FROM AUTHOR]

For general complex or real 1-parameter matrix flows A(t)n, n and for static matrices A ∈ ℂ n , n alike, this paper considers ways to decompose matrix flows and single matrices globally via one constant matrix similarity Cn, n as A(t) = C− 1 ⋅ diag(A1(t),...,Aℓ(t)) ⋅ C or A = C− 1 ⋅diag(A1,...,Aℓ) ⋅ C with each diagonal block Ak(t) or Ak square and their number ℓ exceeding 1 if this is possible. The theory behind our proposed algorithm is elementary and uses the concept of invariant subspaces for the MATLAB eig computed 'eigenvectors' of one associated flow matrix B(ta) to find the coarsest simultaneous block structure for all flow matrices B(tb). The method works efficiently for all time-varying matrix flows A(t), be they real or complex, normal, with Jordan structures or repeated eigenvalues, differentiable, continuous, or discontinuous in t, and likewise for all fixed entry matrices A. Our intended aim is to discover unitarily diagonal-block decomposable flows as they originate in real-time from sensor given data for time-varying matrix problems that are unitarily invariant. Then, the complexities of their numerical treatments decrease by adopting 'divide and conquer' methods for their diagonal blocks. In the process, we discover and study k-normal fixed entry matrix classes that can be decomposed under unitary similarities into various k-dimensional block-diagonal forms. [ABSTRACT FROM AUTHOR]

In this paper, a gradient based iterative method version is proposed for solving the large multilinear systems or tensor equations. A new preconditioner is presented to accelerate the convergence rate of the new iterative methods. Using the linear operator form of the multilinear system, the new methods can be implemented easily for solving several kinds of tensor equations. Finally, some numerical examples are given to illustrate the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

We model the Lights Out game on general simple graphs in the framework of linear algebra over the field F 2 . Based upon a version of the Fredholm alternative, we introduce a separating invariant of the game, i.e., an initial state can be transformed into a final state if and only if the values of the invariant of both states agree. We also investigate certain states with particularly interesting properties. Apart from the classical version of the game, we propose several variants, in particular a version with more than only two states (light on, light off), where the analysis relies on systems of linear equations over the ring Z n . Although it is easy to find a concrete solution of the Lights Out problem, we show that it is NP-hard to find a minimal solution. We also propose electric circuit diagrams to actually realize the Lights Out game. [ABSTRACT FROM AUTHOR]

A recursive formula for computing the exact value of score vectors is proposed for a general form of the linear Gaussian state space model, which is more desirable than approximate values in some statistical analyses. Unlike most extant methods, our formula calculates all components of the score vector simultaneously. This approach significantly simplifies its programing, in particular, with some matrix-oriented programing languages, such as MATLAB. We also consider a way of handling initial conditions that depend on unknown parameters. This issue has not yet been explicitly addressed in the existing literature in the context of exact score computing for a general case, such as the one that we consider in this paper. It is also shown that our formula is especially useful for calculating score tests with an outer product of gradient asymptotic covariance matrix estimator. [ABSTRACT FROM AUTHOR]

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 Jordan normal form. [ABSTRACT FROM AUTHOR]

The nice inversion properties of Toeplitz–Hessenberg matrices from a Hessenbergian representation of Catalan's numbers encourage us to provide explicit inversion formulas (in terms of basic arithmetical operations involving entries from the original matrix) for the non-singular Toeplitz–Hessenberg matrices. Our approach is based on an elementary matrix inflation method, joint with the nested sums formulas. These explicit inversion formulas are then extended to apply to the entries of the inverse of every non-singular properly Hessenberg matrix. [ABSTRACT FROM AUTHOR]

It is shown that a finite-dimensional formally real simple Jordan algebra is completely determined by the relation of Jordan-orthogonality. Communicated by Prof. Alberto Elduque [ABSTRACT FROM AUTHOR]

A general closed interval matrix is a matrix whose entries are closed connected nonempty subsets of R , while an interval matrix is defined to be a matrix whose entries are closed bounded nonempty intervals in R . We say that a matrix A with constant entries is contained in a general closed interval matrix μ if, for every i,j, we have that A i , j ∈ μ i , j . Rohn characterized full-rank square interval matrices, that is, square interval matrices μ such that every constant matrix contained in μ is nonsingular. In this paper, we generalize this result to general closed interval matrices. [ABSTRACT FROM AUTHOR]

An arithmetic based on the multiplication is the base of the Bigeometric (and Multiplicative) Calculus, a type of non-Newtonian Calculus which has had uses in the digital processing of images. In this work, we present the fundamental elements to construct a proposal of Multiplicative Linear Algebra with view of being a support for a Multiplicative Calculus in Several Variables and its applications. Some important concepts such as vector space, linear transformation and matrix representation are reviewed. In particular, we give a meaning and context to the operation: matrix raised to matrix. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the rank-metric codes which are proposed by Delsarte and Gabidulin to be complementary dual codes. We first point out the relationship between Delsarte complementary dual codes (Delsarte LCD codes) and Gabidulin complementary dual codes (Gabidulin LCD codes). We then construct two classes of Gabidulin LCD MRD codes by self-dual basis (or almost self-dual basis) of the finite field F q m over base field F q . Finally, we give an interesting application of rank-metric LCD codes in decoding algorithm. [ABSTRACT FROM AUTHOR]

Subspace codes and constant dimension codes have become a widely investigated research topic due to their significance to error control in random linear network coding. Rank metric codes in Ferrers diagrams can be used to construct good subspace codes and constant dimension codes. In this paper, three constructions of Ferrers diagram rank metric codes are presented. The first two constructions are based on subcodes of maximum rank distance codes, and the last one generates new codes from known Ferrers diagram rank metric codes. Each of these constructions produces optimal codes with different diagrams and parameters for which no optimal construction was known before. [ABSTRACT FROM AUTHOR]

We prove that the commutative nearly absolute valued division algebras have dimension ≤2. This extends a well-known theorem by H. Hopf asserting the same property for each commutative real division algebra of finite dimension. As an application we prove that R , C and C ⋆ are the unique commutative real nearly absolute valued division algebras with norm satisfying ‖ x 2 ‖ = ‖ x ‖ 2 . [ABSTRACT FROM AUTHOR]

The problem of expressing a multivariate polynomial as the determinant of a monic (definite) symmetric or Hermitian linear matrix polynomial (LMP) has drawn a huge amount of attention due to its connection with optimization problems. In this paper we provide a necessary and sufficient condition for the existence of monic Hermitian determinantal representation as well as monic symmetric determinantal representation of size 2 for a given quadratic polynomial. Further we propose a method to construct such a monic determinantal representation (MDR) of size 2 if it exists. It is known that a quadratic polynomial f ( x ) = x T A x + b T x + 1 has a symmetric MDR of size n + 1 if A is negative semidefinite . We prove that if a quadratic polynomial f ( x ) with A which is not negative semidefinite has an MDR of size greater than 2, then it has an MDR of size 2 too. Finally, we characterize all quadratic polynomials that exhibit MDRs of any size. [ABSTRACT FROM AUTHOR]

Consider T ( F)-the ring of all n × n upper triangular matrices defined over some field F. A map φ is called a zero product preserver on T ( F) in both directions if for all x, y ∈ T ( F) the condition xy = 0 is satisfied if and only if φ( x) φ( y) = 0. In the present paper such maps are investigated. The full description of bijective zero product preservers is given. Namely, on the set of the matrices that are invertible, the map φ may act in any bijective way, whereas for the zero divisors and zero matrix one can write φ as a composition of three types of maps. The first of them is a conjugacy, the second one is an automorphism induced by some field automorphism, and the third one transforms every matrix x into a matrix x′ such that { y ∈ T ( F): xy = 0} = { y ∈ T ( F): x′ y = 0}, { y ∈ T ( F): yx = 0} = { y ∈ T ( F): yx′ = 0}. [ABSTRACT FROM AUTHOR]

Considering the natural topology or S -density topology on the domain and on the range we obtain different families of continuous functions f : R → R . We prove that certain differences of those families are strongly c -algebrable, i.e., they contain c -generated free subalgebras. In particular, we obtain strengthenings of recent results of J. Hejduk, A. Loranty, R. Wiertelak, S. Lindner and M. Terepeta. [ABSTRACT FROM AUTHOR]

We construct the definition of allowable graphs of the nonlinear Schrödinger equation of arbitrary degree and use it to verify the separation and irreducibility (over the ring of integers) of the characteristic polynomials of all the possible graphs giving 3-dimensional blocks of the normal form of the nonlinear Schrödinger equation. The method is purely algebraic and the obtained results will be useful in further studies of the nonlinear Schrödinger equation. [ABSTRACT FROM AUTHOR]