Let F be a set of functions with a common domain X and a common range Y. A set S⊆X is called a set of range uniqueness (SRU) for F, if for all f,g∈F, f[S]=g[S]⇒f=g. Let Pn,k be the set of all real polynomials in n variables of degree at most k and let Lk(Rn,Rn) be the set of all linear functions f:Rn→Rn with rank k. We show that there are SRU's for Pn,k of cardinality 2n+kk−1, but there are no such

Two classes of determinants with their matrix entries being polynomial differences are evaluated in closed product expressions. They extend with many free parameters the Vandermonde-like determinant identities associated with the classical root systems.

A complex square matrix is a ray pattern matrix if each of its nonzero entries has modulus 1. A ray pattern matrix naturally corresponds to a weighted-digraph. A ray pattern matrix A is ray nonsingular if for each entry-wise positive matrix K, A∘K is nonsingular. A random model of ray pattern matrices with order n is introduced, where a uniformly random ray pattern matrix B is defined to be the adjacency

Let Fq be a finite field with q elements, n≥2 a positive integer, V0 a n-dimensional vector space over Fq and T0 the set of all linear functionals from V0 to Fq. Let V=V0∖{0} and T=T0∖{0}. The linear functional graph of V0 dented by ϝ(V), is an undirected bipartite graph, whose vertex set V is partitioned into two sets as V=V∪T and two vertices v∈V and f∈T are adjacent if and only if f sends v to the

We investigate the size of the largest entry (in absolute value) in the inverse of certain Vandermonde matrices. More precisely, for every real b>1, let Mb(n) be the maximum of the absolute values of the entries of the inverse of the n×n matrix [bij]0≤i,j

Rays are classes of an equivalence relation on a module V over a supertropical semiring. They provide a version of convex geometry, supported by a ‘supertropical trigonometry’ and compatible with quasilinearity, in which the CS-ratio takes the role of the Cauchy–Schwarz inequality. CS-functions that emerge from the CS-ratio are a useful tool that helps to understand the variety of quasilinear stars

The proper power graph P∗(G) of a group G is the simple graph with non-trivial elements of G as vertices and two distinct elements are adjacent if and only if one is a power of the other. The aim of this paper is to find the structure and to compute the spectrum of the proper power graph of direct product of two finite groups G1 and G2, where at least one of G1 and G2 is an EPO group. Also we provide

In this paper, we focus on the extensions of the Bessel matrix function and the modified Bessel matrix function. We first introduce the extended Bessel matrix function and the extended modified Bessel matrix function of the first kind by using the extended Beta matrix function. Then we establish the integral representations, differentiation formula, and hypergeometric representation of such functions

Recently, duality principles in g-frame theory were studied. This paper addresses a relaxation of g-R-duals. We introduce the notion of weak g-R-dual in g-frame theory setting, and present the link between frame properties of a bounded operator sequence and its weak g-R-duals. Using weak g-R-duals, we characterize g-frames and (unitary) equivalence between g-frames. And using pseudo-inverse operators

ABSTRACT In this paper, we define two new concepts for a pair of complex square matrices of the index at most 1, the core orthogonal pair and strongly core orthogonal pair. The relationship between core orthogonality (resp. strong core orthogonality) and associated projectors of a pair of matrices is established. Using the core commutativity of two matrices of the index at most 1, a characterization

Let k,n1,…,nk be positive integers such that ni⩾2 for i=1,…,k and let Mni denote the algebra of ni×ni matrices over a field F for i=1,…,k. Let ⨂i=1kMni be the tensor product of Mn1,…,Mnk. We obtain a structural characterization of additive maps ψ:⨂i=1kMni→⨂i=1kMni satisfying ψ⨂i=1kAi⨂i=1kAi=⨂i=1kAiψ⨂i=1kAifor all A1∈Sn1,…,Ak∈Snk, where Sni=Est(ni)+αEpq(ni):α∈F,1⩽p,q,s,t⩽ni are not all distinct integersEpq(ni)and

Let R be an associative and commutative ring with unity 1 and consider k∈N such that 1+1+⋯+1=k is invertible. Let UT∞(k)(R) be the group of upper triangular infinite matrices whose diagonal entries are kth roots of 1. We show that every element of the group UT∞(R) can be expressed as a product of 4k−6 commutators all depending on the powers of elements in UT∞(k)(R) of order k. If R is the complex number

In this paper we give an in-deph analysis of the nilpotency index of nilpotent homogeneous inner superderivations in associative prime superalgebras with and without superinvolution. We also present examples of all the different cases that our analysis exhibits for the nilpotency indices of the inner superderivations.

Let (M,τ) be a semi-finite von Neumann algebra, L0(M) be the set of all τ-measurable operators and μt(x) be the generalized singular number of x∈L0(M). We proved that if 10 holds.

We introduce and investigate the orbit-closed C-numerical range, a natural modification of the C-numerical range of an operator introduced for C trace-class by Dirr and vom Ende. Our orbit-closed C-numerical range is a conservative modification of theirs because these two sets have the same closure and even coincide when C is finite rank. Since Dirr and vom Ende's results concerning the C-numerical

In this paper, we first study zero divisor graphs over finite chain rings. We determine their rank, determinant, and eigenvalues using reduction graphs. Moreover, we extend the work to zero divisor graphs over finite commutative principal ideal rings using a combinatorial method, finding the number of positive eigenvalues and the number of negative eigenvalues, and finding upper and lower bounds for

A strongly connected digraph is called a cactoid-type digraph if each of its blocks is a digraph consisting of finitely many oriented cycles sharing a common directed path. In this article, we find the formula for the determinant of the distance matrix for a weighted cactoid-type digraph and find its inverse, whenever it exists. We also compute the determinant of the distance matrix for a class of

A generalized tournament matrix M is a nonnegative matrix that satisfies M+Mt=J−I, where J is the all ones matrix and I is the identity matrix. In this paper, a characterization of generalized tournament matrices with the same principal minors of orders 2, 3, and 4 is given. In particular, it is proven that the principal minors of orders 2, 3, and 4 determine the rest of the principal minors.

Let F be a field of characteristic 0, E be the unitary infinite dimensional Grassmann algebra over F and consider the algebra M1,1E with its natural Z2-grading. We describe the graded A-identities for M1,1E and we compute its graded A-codimensions.

Over a general field and for an arbitrary positive integer k, those triangular matrices whose kth power is diagonal are explicitly characterized. This is then used to characterize those complex matrices whose kth power is normal. This gives corresponding results for matrices whose kth power is Hermitian or real symmetric. Our results generalize and imply a recent result about eventually normal and

In this paper, we extend the notion of banded matrices to matrices where two bands are allowed, called double banded matrices. Our main aim is to establish the periodicity of the determinants for (0,1) double banded matrices. As a corollary, we answer to two recent conjectures and other extensions. Several illustrative examples are provided as well.

ABSTRACT Let A1,…,As be unitary commutative rings which do not have non-trivial idempotents and let A=A1⊕⋯⊕As be their direct sum. We describe all idempotents in the 2×2 matrix ring M2(A[[X]]) over the ring A[[X]] of formal power series with coefficients in A and in an arbitrary set of variables X. We apply this result to the matrix ring M2(Zn[[X]]) over the ring Zn[[X]] where Zn≅Z/nZ for an arbitrary

In this paper, we firstly establish new numerical radius inequalities which refine a result of Kittaneh in [Studia Math. 168, 73–80 (2005)], then present some numerical radius inequalities involving non-negative increasing convex functions for n×n operator matrices, which generalize the related results of Shebrawi in [Linear Algebra Appl. 523(15), 1–12 (2017)].

Brualdi and Hoffman [On the spectral radius of (0, 1)-matrices. Linear Algebra Appl. 1985;65:133–146] proposed the problem of determining the maximal spectral radius of graphs with a given size. In this paper, we consider the Brualdi–Hoffman-type problem for graphs with a given matching number. The maximal Q-spectral radius of graphs with a given size and matching number is obtained, and the corresponding

We study the connection between probability distributions satisfying certain conditional independence (CI) constraints, and point and line arrangements in incidence geometry. To a family of CI statements, we associate a polynomial ideal whose algebraic invariants are encoded in a hypergraph. The primary decompositions of these ideals give a characterization of the distributions satisfying the original

We give a simple proof of the existence of a complex eigenvalue of real or complex square matrices with Linear Algebra accessible to students beginning a first course on the subject, namely using just the Cauchy–Schwarz inequality in Cn, linear independence and the matrix product, together with the Weierstrass extreme value theorem. It yields an equally simple proof of the Fundamental Theorem of Algebra

The aim of this work is to show the consistency of all systems of non-homogeneous linear difference equations of the form φ(xn+1)=xn+v0, where φ∈Endk⁡(V) is a finite potent endomorphism of an arbitrary vector space V and v0∈V. An algorithm to compute the set of solutions of these systems is given. In particular, the method offered is valid for computing the explicit solutions of the system of non-homogeneous

We characterize extreme contractions defined between finite-dimensional polyhedral Banach spaces using k-smoothness of operators. As application of results obtained, we explicitly compute the number of extreme contractions in some special Banach spaces. Our approach, in this paper, in studying extreme contractions lead to the improvement and generalization of previously known results.

In this paper, we obtain some inequalities for the Berezin number of operators on reproducing kernel Hilbert spaces by using the Krein–Lin inequality and refinements of the Young inequality. Also, we give upper bounds for ber2(A)−ber(A2) on reproducing kernel Hilbert spaces.

Let Gn,β be the set of simple graphs of order n with given matching number β. In this paper, we characterize the extremal graphs with maximal Aα-spectral radius in Gn,β for 0≤α<1, which generalizes the results for adjacency matrix in [Feng L, Yu G, Zhang X-D. Spectral radius of graphs with given matching number. Linear Algebra Appl. 2007;422:133–138, Theorem 1.1] and signless Laplacian matrix in [Yu

We define a weighted alternating zeta function of a digraph D and give its determinant expression. We present a decomposition formula for the weighted alternating zeta function of a group covering of D. Furthermore, we introduce a weighted alternating L-function of D and present a determinant expression of it. As a corollary, we present a decomposition formula for the weighted alternating zeta function

ABSTRACT Let G be a graph with order n and size m. For each real number α∈[0,1], the Aα-matrix of a graph G is defined as Aα(G):=αD(G)+(1−α)A(G), where D(G)=diag(d1,…,dn) is the degree matrix of G, and A(G) is its adjacency matrix. The functions s(G)=∑1≤i≤ndi−2mnandvar(G)=1n∑1≤i≤ndi−2mn2have been already used elsewhere as measures of graph irregularity. In this paper, we find several bounds for the

This work examines different notions of similarity for a pair u, v of vertices of a graph G. The considered similarity notions are either of removal type or attachment type. In the removal case, checking similarity between u and v is a matter of comparing the vertex-deleted subgraphs G−u and G−v according to a certain criterion (isomorphism, degree sequence, index, characteristic polynomial, complementarity

ABSTRACT We present characterizations and representations for the CMP inverse. Also, we explore the CMP inverse of a block triangular matrix and its sign pattern, propose a successive matrix squaring algorithm for computing the CMP inverse, derive perturbation bounds and the continuity of the CMP inverse.

ABSTRACT In Markov chain theory, stochastic matrices are used to describe inter-state transitions. Powers of such transition matrices are computed to determine the behaviour within a Markov system. For this, diagonalizable matrices are preferred because of their useful properties. The non-diagonalizable matrices are therefore undesirable. The aim is to determine a nearby diagonalizable matrix A~, starting

In this paper, we prove the fundamental theorem of Hopf modules associated with a Hopf brace.

Let Ωn denote the class of n×n doubly stochastic matrices (each such matrix is entrywise nonnegative and every row and column sum is 1). We study the diagonals of matrices in Ωn. The main question is which A∈Ωn are such that the diagonals in A that avoid the zeros of A all have the same sum of their entries. We give a characterization of such matrices and establish several classes of patterns of such

We present two generalizations of Singular Value Decomposition from real-numbered matrices to dual-numbered matrices. We prove that every dual-numbered matrix has both types of SVD. Both of our generalizations are motivated by applications, either to geometry or to mechanics.

ABSTRACT In this work, we explore an extension of the Omega calculus in the context of matrix analysis introduced recently by Neto [Matrix analysis and Omega calculus. SIAM Rev. 2020;62(1):264–280]. We obtain Omega representations of analytic functions of three important classes of matrices: companion, tridiagonal, and triangular. Our representation recovers the main results of Chen and Louck [The

As solutions of some operator equations, we present two new outer inverses for bounded linear operators between two Hilbert spaces. Precisely, we introduce and characterize the weighted weak core inverse and the dual weighted weak core inverse. These generalized inverses are extensions of the weak core inverse and its dual which have been recently introduced for complex square matrices. Also, we give

Let 0→a→e→g→0 be an abelian extension of the Lie superalgebra g. In this article we consider the problems of extending endomorphisms of a and lifting endomorphisms of g to certain endomorphisms of e. We connect these problems to the cohomology of g with coefficients in a through construction of two exact sequences which are our main results. The first exact sequence is obtained using the Hochschild–Serre

If A,B∈B(H) are normal accretive operators, X∈B(H),0<α<1 and Φ is a s.n. function, we proved that |(A∗+A)1−αX(B+B∗)α|2⩽Γ(2−2α)∫[0,+∞)e−tB∗(B∗+B)α|AX+XB|2×(B∗+B)αe−tBt2α−1dt, ∥(A∗+A)1−αX(B+B∗)α∥Φ⩽Γ(2−2α)Γ(2α)∥AX+XB∥Φ,if AX+XB∈CΦ(H).Let A,B,X∈B(H),A⩾0,B⩾0,η,θ∈R,α∈(0,1) and Φ be a s.n. function. If eiηAX+eiθXB∈CΦ(H), then we have the following generalization of Young's norm inequality in Jocić [Cauchy–Schwarz

ABSTRACT Let B(H) denote the algebra of all bounded linear operators acting on a complex Hilbert space H. For A,B∈B(H), define the bimultiplication operator M2,A,B on the class of Hilbert–Schmidt operators by M2,A,B(X)=AXB. In this paper, we show that if B is normal, then co(W0(A)W0(B))⊆W0(M2,A,B),where co stands for the convex hull and W0(.) denotes the maximal numerical range. If in addition, A is

A new geometric proof of a known result characterizing the quaternionic numerical range of normal matrices is proposed. Our proof can be interpreted in probabilistic terms.

Using interpolation classes and the generalized Petz's trace inequality we give a new matrix inequality which might play an important role in the quantum information theory: For any pair of positive definite matrices A,B∈Mn(C) we have Tr⁡((A−B)(exp⁡(A)−exp⁡(B))≤Tr⁡(|A(exp⁡(A)−1)−B(exp⁡(B)−1)|).

In this paper, we propose generalized inverse power methods with variable shifts for finding the smallest H-/Z-eigenvalue and associated H-/Z-eigenvector of symmetric tensors. The methods are guaranteed to always converge to a H-/Z-eigenpair. Furthermore, for an even order nonsingular symmetric M-tensor, the proposed method with any positive initial point always converges to the smallest H-eigenvalue

The class of polynomially normal operators is a wider class than the class of all normal operators. Inspired by some interesting well known facts about normal operators and by some recent work, we present new properties of polynomially normal operators. Precisely, we prove that under certain conditions polynomially normal operators are Drazin or even group invertible and we also give necessary and

The Popoviciu inequality is known as a weighted version of the celebrated Aczel inequality. In this paper, we establish an analogue of the Popoviciu inequality for the determinants of positive definite matrices, and then we extend our result to the case where the underlying matrices are sectorial matrices, i.e. matrices whose field of values (or numerical range) are contained in a sector.

The study of m-tuples of operators was the subject of intensive research by many authors. Our aim in this work is to consider a generalization of the concept of posinormality of a single operator done in Rhaly (Posinormal operators. J Math Soc Jpn. 1994;46(4):587–605) to the concept of joint posinormality for m-tuples of operators on a complex Hilbert space. We study some of the basic properties of

Let D be a simple digraph with eigenvalues z1,z2,…,zn. The energy of D is defined as E(D)=∑i=1n|Re(zi)|, where Re(zi) is the real part of the eigenvalue zi. In this paper, a lower bound for the spectral radius of D will be established based on the number of subgraphs P3↔ in D, improving some of the lower bounds that appear in the literature. Furthermore, this result allows us to obtain an upper bound

Let N be the algebra of all n×n dominant block upper triangular matrices over a field. In this paper, we explicitly describe all linear Lie centralizers of N. We also describe linear Lie centralizers of the algebra B of block upper triangular matrices over a field.

A tensor v is the sum of at least rank⁡(v) elementary tensors. In addition, a ‘border rank’ is defined: rank_(w)=r holds if r is the minimum integer such that w is a limit of rank-r tensors. Usually, the set of rank-r tensors is not closed, i.e. tensors with r=rank_(w)

Let R be a unitary prime ring with the maximal left ring of quotients Qml(R). We completely characterize the forms of additive maps F1,F2,G1,G2:R→Qml(R) such that F1(x)y+F2(y)x+xG1(y)+yG2(x)=0whenever x,y∈R satisfy xy = 0 = yx under the assumptions that R contains a nontrivial idempotent and characteristic of R is not 2. This partially generalizes a result of T.K. Lee in [Bi-additive maps of ζ-Lie

(2021). Preface of the special issue on Numerical ranges and numerical radii. Linear and Multilinear Algebra: Vol. 69, Numerical Ranges and Numerical Radii, pp. 771-771.

In the last decade, several scholars proposed an unifying approach to study the spectral theories of the adjacency, Laplacian and signless Laplacian of graphs. The most general graph matrix is the universal adjacency matrix U=αA+βD+γJ+δI, where A, D, J, and I are the adjacency matrix of G, the degree matrix of G, the all-ones matrix, the identity matrix, respectively. Here, we consider Mβ=A+βD, with

The distance Laplacian matrix of a connected graph G is defined as L(G)=Tr(G)−D(G), where Tr(G) is the diagonal matrix of the vertex transmissions in G and D(G) is the distance matrix of G. The largest eigenvalue of L(G) is called the distance Laplacian spectral radius of G. In this paper, we determine the graphs with the maximum distance Laplacian spectral radius and the minimum distance Laplacian

ABSTRACT A hypergraph is said to be oriented if each edge-vertex incidence has a label of +1 or −1. An oriented hypergraph is called incidence balanced if there exists a bipartition of the vertex set such that every edge intersects one part of the bipartition in positively incident vertices with the edge and the other part in negatively incident vertices with the edge. In this paper, we investigate

We study smooth quadric surfaces in the Pfaffian hypersurface in parameterizing skew-symmetric matrices of rank at most 4, not intersecting its singular locus. Such surfaces correspond to quadratic systems of skew-symmetric matrices of size 6 and constant rank 4, and give rise to a globally generated vector bundle E on the quadric. We analyse these bundles and their geometry, relating them to linear

Let H be the twisted Heisenberg–Virasoro algebra. In this paper, we first present a method to produce a class of tensor product H-modules, by which we can obtain the known weight H-modules M(V,Aα,b) and non-weight H-modules M(V,Ω(λ)). Then we study a class of linear tensor product non-weight modules M(V,Ω(λ0))⊗⨂i=1mΩ(λi,αi,βi) over H. The necessary and sufficient conditions for M(V,Ω(λ0))⊗⨂i=1mΩ(λi

We study the behaviour of contact brackets on the tensor product of two algebras, in particular, address the question of Martínez and Zelmanov about extension of a contact bracket on the tensor product from the brackets on the factors.