Let D be an infinite division ring with centre F and T(∞,D) the group of infinite upper triangular invertible matrices indexed by N×N over D. In this paper, we first show that if D is finite dimensional over F, then every element in T(∞,D) whose diagonal entries are commutators of D∖{0} is a commutator of an infinite diagonal matrix and another infinite upper triangular matrix. Some applications are

In this paper, first we consider cross-diagonal, in particular off-diagonal, operator matrices; those operator matrices such that all entries except those on the main and off diagonals are zero. We show that this operator matrix is unitarily equivalent to a block diagonal operator matrix whose diagonal blocks are all two-by-two, except at most one of them which is one-by-one. Then, using this unitary

This paper is mainly concerned with a model updating problem for damped structural systems, which can be described as the following inverse quadratic eigenvalue problem (IQEP) and an optimal approximate problem (OAP). Problem IQEP: Given matrices Λ=diag{λ1,…,λp}∈Cp×p and X=[x1,…,xp]∈Cn×p with p≤n, λi≠λj for i≠j, i,j=1,…,p, rank(X)=p and both Λ and X being closed under complex conjugation, find real

The problem of determining lower bounds for the second and numerically smallest eigenvalue of A∘A#, when A is a group invertible singular M-matrix, is considered.

Let G be a graph. The energy of G, E(G), is defined as the sum of absolute values of its eigenvalues. Here, it is shown that if G is a graph and {H1,…,Hk} is an edge partition of G, such that H1,…,Hk are spanning; then E(G)=∑i=1kE(Hi) if and only if AiAj=0, for every 1⩽i,j⩽k and i≠j, where Ai is the adjacency matrix of Hi. It was proved that if G is a graph and H1,…,Hk are subgraphs of G which partition

In this paper, an algorithm is proposed to compute the inverse of an invertible matrix. The new algorithm is a generalization of the algorithms based on the well-known Schultz-type iterative methods. We show that the convergence order of the new method is a linear combination of the Fibonacci sequence and also is powerful and efficient in finding and keeping sparsity of the obtained approximate inverse

Let A and B be positive definite matrices of the same size, and let α≥1. If the spectrum of AB−1 lies in the interval [1,α], then |||AtB1−t+BtA1−t|||≤α|||A+B|||for t∈[0,1] and for every unitarily invariant norm. This norm inequality is closely related to an open question of Bourin.

Given a totally positive matrix, can one insert a line (row or column) between two given lines while maintaining total positivity? This question was first posed and solved by Johnson and Smith who gave an algorithm that results in one possible line insertion. In this work, we revisit this problem. First, we show that every totally positive matrix can be associated with a certain vertex-weighted graph

Using some classical theorems on entire functions, we determine the asymptotic for the counting function of the resonances for 2×2 matrix Schrödinger operator in one dimension for potentials with compact support.

This paper deals with a new almost convergent sequence space constructed by Schröder matrix. It is shown that this new sequence space is linearly isomorphic to the space of all almost convergent sequences. Also, the β-dual of the resulting space is expressed and Schröder core of a complex-valued sequence is introduced. Moreover, some inclusion theorems are obtained for this new type of core.

In this paper, we give a method for the evaluation of Hankel determinant of a linear combination of Catalan numbers. Then, we extend this result to shifted Catalan numbers. This is done by finding a Jacobi linear functional, such that their moments are the Catalan numbers or the shifted Catalan numbers. The values of such determinants are then expressed in terms of Jacobi polynomials.

New bounds for the Perron root ρ(A) of a positive matrix A are proposed. Let  R1≥R2≥⋯≥Rn be the row sums of matrix A listed in descending order. We use a pairwise combination method to prove that min1≤k≤j{(1−t)Rk+tRn−k+1}≤ρ(A)≤max1≤k≤j{tRk+(1−t)Rn−k+1},with j=⌈n/2⌉, t=M/(M+m), and M (respectively m) is the maximum (resp. minimum) entry of A. This result improves the well-known bounds Rn≤ρ(A)≤R1, and

In this work, some new upper and lower bounds for the Davis–Wielandt radius are introduced. Generalizations of some presented results are obtained. Some bounds for the Davis–Wielandt radius for n×n operator matrices are established. An extension of the Davis–Wielandt radius to the Euclidean operator radius is introduced.

The matching number of G, written α′(G), is the size of a maximum matching in G. Suppose that n and k are positive integers of the same parity. Let θ′(n,k) be the largest root of x3−(n−k−2)x2−(n−1)x+k(n−k−2)=0 and θ(n,k)={θ′(n,k)ifn≥3k+2n−k−2+(n−k−2)2+4(n2−k2)4ifn≤3k.In this article, we prove that for a positive integer n≥k+2, if G is an n-vertex connected graph with the spectral radius ρ(G)>θ(n,k)

We present a generalization of the pseudoinverse operation to pairs of matrices, as opposed to single matrices alone. We note the fact that the Singular Value Decomposition can be used to compute the ordinary Moore-Penrose pseudoinverse. We present an analogue of the Singular Value Decomposition for pairs of matrices, which we show is inadequate for our purposes. We then present a more sophisticated

A d-dimensional tensor A of format n×n×⋯×n defines naturally a rational map Ψ from the projective space Pn−1 to itself and its eigenscheme is then the subscheme of Pn−1 of fixed points of Ψ. The eigendiscriminant is an irreducible polynomial in the coefficients of A that vanishes for a given tensor if and only if its eigenscheme is singular. In this paper, we contribute two formulas for the computation

The question of which elements of the open unit disc occur as eigenvalues of n-by-n doubly stochastic matrices has long been open, in spite of a number of intriguing partial results. By enhancing a natural, but slightly false, conjecture, we gain some new computational insights into the problem. We also apply the classical field of values to give some partial results on both the necessity of certain

Let L=(L1,L2) be a list consisting of structural data for a matrix polynomial; here L1 is a sublist consisting of powers of irreducible (monic) scalar polynomials over the field R, and L2 is a sublist of nonnegative integers. For an arbitrary such L, we give easy-to-check necessary and sufficient conditions for L to be the list of elementary divisors and minimal indices of some real T-palindromic quadratic

Given a matrix A∈Cn×n there exists a nonsingular matrix V such that V−1AV=J, where J is a very sparse matrix with a diagonal block structure, known as the Jordan canonical form (JCF) of A. Assume that A is nonsingular and that V and J are given. How to obtain V^ and J^ such that V^−1A−1V^=J^ and J^ is the JCF of A−1? Curiously, the answer involves the Pascal matrix. For the Frobenius canonical form

The lengths of matrix incidence algebras are studied when their radicals have square zero. All realizable values of the length function are provided for such algebras. In order to obtain this result, a discrete optimization problem is posed and solved. Also, the exact formula of the length is deduced under the additional assumption that the algebra is maximal by inclusion. Moreover, the solution to

In this article, we define a set of matrices analogous to Vaserstein-type matrices, which were introduced in the paper ‘Serre's problem on projective modules over polynomial rings and algebraic K-theory’ by Suslin–Vaserstein in 1976. We prove that these are elementary linear matrices. Also, under some conditions, these matrices belong to Petrov's odd unitary group, which is a generalization of all

Let G be a connected graph. The principal ratio of G is the ratio of the maximum and minimum entries of its Perron eigenvector. In 2007, Cioabă and Gregory conjectured that among all connected graphs on n vertices, the kite graph attains the maximum principal ratio. In 2018, Tait and Tobin confirmed the conjecture for sufficiently large n. In this article, we show the conjecture is true for all n≥5000

Let A be a d×d matrix with rational entries which has no eigenvalue λ∈C of absolute value |λ|<1 and let Zd[A] be the smallest nontrivial A-invariant Z-module. We lay down a theoretical framework for the construction of digit systems (A,D), where D⊂Zd[A] finite, that admit finite expansions of the form x=d0+Ad1+⋯+Aℓ−1dℓ−1(ℓ∈N,d0,…,dℓ−1∈D)for every element x∈Zd[A]. We put special emphasis on the explicit

It is shown that, in any semigroup with involution, both the DMP inverse of Malik and Thome [Appl Math Comp. 2014;226:575–580.] and the CMP inverse of Mehdipour and Salemi [Linear Multilinear Alg. 2018;66:1046–1053.] are special cases of the Bott–Duffin (e,f)-inverse introduced in [Linear Algebra Appl. 2012;436:1909-1923.] It is also shown that the core inverse of O.M. Baksalary and Trenkler [Linear

We generalize a two-parameter extended Wigner–Yanase skew information given by Z. Zhang [J Math Anal Appl. 2021;497(1): 124851] to any general operator monotone function. We prove several fundamental properties including the joint concavity and an uncertainty relation for the proposed generalized Wigner–Yanase skew information.

Let Gn,β be the set of graphs of order n with given matching number β. Let D(G) be the diagonal matrix of the degrees of the graph G and A(G) be the adjacency matrix of the graph G. The largest eigenvalue of the nonnegative matrix Aα(G)=αD(G)+A(G) is called the α-spectral radius of G. The graphs with maximal α-spectral radius in Gn,β are completely characterized in this paper. In this way, we provide

This paper studies two inverse eigenvalue problems for two kinds of acyclic matrices whose graphs are caterpillars. The spectral data of the first problem considers the minimal and maximal eigenvalues of all leading principal submatrices of the matrix. The second consists of an extremal eigenvalue of each leading principal submatrix and one eigenpair of the matrix. In the main results, we give sufficient

For a subspace S of Cn and a fixed basis, we study the compact and convex set mS=convexhull {|s|2∈R≥0n:s∈S and ‖s‖=1}≃{Diag(Y)∈Mnh(C):Y≥0,tr(Y)=1,PSYPS=Y}that we call the moment of S, where |s|2=(|s1|2,|s2|2,…,|sn|2). This set is relevant in the determination of minimal hermitian matrices ( M∈Mnh such that ‖M+D‖≤D for every diagonal D and the spectral norm ‖⋅‖). We describe extremal points and certain

In this paper, we propose a modified generalized SOR-like (MGSOR) method for solving an absolute value equation (AVE), which is obtained by reformulating equivalently AVE as a two-by-two block nonlinear equation and by introducing the transformation Py:=|x| with a general nonsingular matrix P. The convergence results of the MGSOR method are obtained under certain assumptions imposed on the involved

Let C be a conjugation on a complex separable Hilbert space H. A bounded linear operator T is said to be C-normal if CT∗TC=TT∗. In this paper, first, we give a representation of C-normal operators on finite dimensional Hilbert space and later extend it to compact C-normal operators on infinite-dimensional separable Hilbert spaces. In the end, we discuss the eigenvalue problem for C-normal operators

A number of differences exist between the structures of Hilbert spaces and Hilbert C∗-modules. This work aims at studying some properties and characterizations of c-K-g-frames in Hilbert C∗-modules. Moreover, the dual c-K-g-Bessel system for a given c-K-g-frame in a Hilbert C∗-module is introduced and all dual c-K-g-Bessel systems are characterized. Finally, we provide sufficient conditions under which

In this paper, we introduce the notion of (A,m)-partial isometry for a positive operator A and a nonnegative integer m. This family of operators contains both the class of (A,m)-isometries discussed in Sid Ahmed and Saddi [A-m-isometric operators in semi-Hilbertian spaces. Linear Algebra Appl. 2012;436:3930–3942] and that of m-partial isometries introduced in Saddi and Sid Ahmed [m-partial isometries

In this paper, we consider the contractive real symmetric matrix completion problems motivated in part by studies on sparse (or dense) matrices for weighted sparse recovery problems and rating matrices with rating density in recommender systems. We completely characterize symmetric patterns P with the property (C) that every partially contractive real symmetric matrix with pattern P has a contractive

A complex unit gain graph is a triple Φ=(G,T,φ) (or Gφ for short) consisting of a simple graph G, as the underlying graph of Gφ, the set of unit complex numbers T={z∈C:|z|=1} and a gain function φ:E→→T such that φ(ei,j)=φ(ej,i)−1. Let A(Gφ) be the adjacency matrix of Gφ. In this paper, we prove that m(G)−c(G)≤p(Gφ)≤m(G)+c(G),m(G)−c(G)≤n(Gφ)≤m(G)+c(G),where p(Gφ), n(Gφ), m(G) and c(G) are the number

Inspired by natural classes of examples, we define generalized directed semi-trees and construct weighted shifts on them. Given an n-tuple of generalized directed semi-trees with certain properties, we associate an n-tuple of multiplication operators on a Hilbert space H2(β) of formal power series. Under certain conditions, H2(β) turns out to be a reproducing kernel Hilbert space consisting of holomorphic

In this article, we treat the space H of real quaternions as a Lie algebra equipped with its commutator product. We show that all involutions of this Lie algebra that are automorphisms (respectively, anti-automorphisms) and restrict to the identity on the centre R⋅1 (sometimes called automorphisms of the first kind) are actually algebra automorphisms (resp. anti-automorphisms) of the division algebra

We discover a geometric property of the space of tensors of fixed multilinear (Tucker) rank. Namely, it is shown that real tensors of the fixed multilinear rank form a minimal submanifold of the Euclidean space of tensors endowed with the Frobenius inner product. We also establish the absence of local extrema for linear functionals restricted to the submanifold of rank-one tensors, finding application

This paper introduces a modified block product preconditioner for a class of complex symmetric linear systems which is constructed by the equivalent real block two-by-two structure of the coefficient matrix. We discuss the convergence of the associated iterative method and determine the optimal choice of the iteration parameter. Furthermore, some algebraic properties of the preconditioned matrix are

Let k be an algebraically closed field of characteristic different from 2. Up to isomorphism, the algebra Matn×n⁡(k) can be endowed with a k-linear involution in one way if n is odd and in two ways if n is even. In this paper, we consider r-tuples A∙∈Matn×n⁡(k)r such that the entries of A∙ fail to generate Matn×n⁡(k) as an algebra with involution. We show that the locus of such r-tuples forms a closed

Let R be a prime ring of characteristic different from 2, Qr its right Martindale quotient ring and C its extended centroid. Suppose that F is a generalized skew derivation of R, L a non-central Lie ideal of R, a∈R a non-zero element of R, m,r≥1 and s,t≥0 fixed integers. If a(F(u)r(F(u)su+uF(u)s)F(u)t)m=0 for all u∈L, then either there exists b∈Qr such that F(x)=bx, for any x∈R, with ab = 0 or R⊆M2(K)

The class of unicyclic non-bipartite graphs with unique perfect matching, denoted by U, is considered in this article. This article provides a complete characterization of unicyclic graphs in U which possess tricyclic inverses.

In this paper, the author gives a description of the derivations of UTMn(R), the ring of upper triangular matrices over an associative ring R with identity. The main result states that if D is an arbitrary derivation of the ring UTMn(R) and A∈UTMn(R), then there are matrices, such that the derivative D(A) is a linear combination of the values of well-known derivations of these matrices.

In this paper, we introduce operator geodesically convex and operator convex-log functions and characterize some properties of them. Then we apply these classes of functions to present several operator Azcél and Minkowski-type inequalities extending some known results. The concavity counterparts are also considered.

In this paper, we introduce a new conservative matrix involving Schröder numbers and investigate its matrix domain in the sequence spaces c and c0. Furthermore, the Köthe–Toeplitz dual, generalized Köthe–Toeplitz dual and Garling dual are expressed and several matrix transformations are given based on the new Banach spaces.

The common characterizations and various individual properties of different generalized inverses are established. Several equivalent conditions for the core-EP, weak group inverse, m-weak group inverse and weak core inverse are presented. The brand new explicit expressions for the operator binary relations defined by various generalized inverses are obtained. We derive properties and study the relationship

We give a constructive sufficient condition for a matrix over a commutative ring to be von Neumann regular, and we show that it is also necessary over local rings. Specifically, we prove that a matrix A over a local commutative ring is von Neumann regular if and only if A has an invertible ρ(A)×ρ(A)-submatrix if and only if the determinantal rank ρ(A) and the McCoy rank of A coincide. We deduce consequences

Let E be the Grassmann algebra of an infinite-dimensional vector space L over a field of characteristic zero. In this paper, we study the Z-gradings on E having the form E=E(r1,r2,r3)(v1,v2,v3), in which each element of a basis of L has Z-degree r1,r2, or r3. We provide a criterion for the support of this structure to coincide with a subgroup of the group Z, and we describe the graded identities for

The purpose of this paper is to establish new log-majorization results concerning eigenvalues and singular values which generalize some previous work related to a conjecture and an open question which were presented by Lemos and Soares in 2018. In addition, we present a complement of a unitarily invariant norm inequality which was conjectured by Bhatia, Lim and Yamazaki in 2016, and recently proved

Let A and B be unital rings. An additive map T:A→B is called a weighted Jordan homomorphism if c=T(1) is an invertible central element and cT(x2)=T(x)2 for all x∈A. We provide assumptions, which are in particular fulfilled when A=B=Mn(R) with n≥2 and R any unital ring with 12, under which every surjective additive map T:A→B with the property that T(x)T(y)+T(y)T(x)=0 whenever xy = yx = 0 is a weighted

Let H and K be complex infinite dimensional separable Hilbert spaces. For given operators A∈B(H) and B∈B(K), we provide necessary and sufficient conditions which make a 2×2 upper triangular operator matrix MC=(AC0B) a CFI operator for some (or every) C∈B(K,H).

The origins of the notion of matchings in groups spawn from a linear algebra problem proposed by E. K. Wakeford [On canonical forms. Proc London Math Soc (2). 1920;18:403–410] which was tackled in Fan and Losonczy [Matchings and canonical forms for symmetric tensors. Adv Math. 1996;117(2):228–238]. In this paper, we first discuss unmatchable subsets in abelian groups. Then we formulate and prove linear

In this paper, we generalize several inequalities involving powers of the Davis-Wielandt radius for off-diagonal part of 2×2 operator matrices of the form T=(0BC0), where B and C are two bounded linear operators. We also obtain some new upper and lower bounds for the Davis-Wielandt radius of bounded linear operators, which generalize the existing ones.

We introduce Poisson triple systems, which are vector spaces with 3 trilinear operations satisfying 9 polynomial identities of degree 5. We show that every Poisson triple system has a universal enveloping Poisson algebra. Finally, we briefly discuss operadic aspects of Poisson triple systems.

ABSTRACT In this paper, we study the identifiability of specific forms (symmetric tensors), with the target of extending recent methods for the case of 3 variables to more general cases. In particular, we focus on forms of degree 4 in 5 variables. By means of tools coming from classical algebraic geometry, such as Hilbert function, liaison procedure and Serre's construction, we give a complete geometric

The paper deals with the generalized numerical radius of linear operators acting on a complex Hilbert space H, which are bounded with respect to the seminorm induced by a positive operator A on H. Here A is not assumed to be invertible. Mainly, if we denote by ωA(⋅) and ω(⋅) the generalized and the classical numerical radii respectively, we prove that for every A-bounded operator T we have ωA(T)=ω(A1/2T(A1/2)†)

We introduce a natural notion of determinants in matrix JB ∗-algebras, i.e. for hermitian matrices of biquaternions and for hermitian 3×3 matrices of complex octonions. We establish several properties of these determinants which are useful to understand the structure of the Cartan factor of type 6. As a tool, we provide an explicit description of minimal projections in the Cartan factor of type 6 and

A c-cyclic graph is a connected graph with n vertices and n + c−1 edges. In this paper, we consider the problem that: in the class of graphs, each of which is the complement graph of a c-cyclic graph with n vertices, which graph has the largest Aα-spectral radius. We shall show that, for 0≤c≤n−4, the extremal graph must be a graph with an isolated vertex. This implies that the maximum degree of G is

For a tree T, U(T) denotes the minimum number of eigenvalues of multiplicity 1 among all real symmetric matrices, whose graph is T. It is known that U(T)≥2. A tree is linear if all its vertices of degree at least 3 lie on a single induced path, and k-linear if there are k of these high degree vertices. U(T) is known for generalized stars (1-linear trees), and U(T) is determined here for 2-linear trees

Let g be a finite-dimensional complex Lie algebra and HLiem(g) be the affine variety of all multiplicative Hom-Lie algebras on g. We use a method of computational ideal theory to describe HLiem(gln(C)), showing that HLiem(gl2(C)) consists of two 1-dimensional and one 3-dimensional irreducible components and HLiem(gln(C))={diag{δ,…,δ,a}∣δ=1or0,a∈C} for n⩾3. We construct a new family of multiplicative