A T -gain graph is a triple Φ = ( G , T , ϕ ) consisting of an underlying graph G = ( V ( G ) , E ( G ) ) , the circle group T = { z ∈ C : | z | = 1 } and a gain function ϕ : E → → T , such that ϕ ( e i j ) = ϕ ( e j i ) − 1 = ϕ ( e j i ) ¯ . In this paper, we focus our attention on the relations between the inertia indices of T -gain graph Φ and the inertia indices of its underlying graph G. We obtain

We give a lower bound for the numerical index of two-dimensional real spaces with absolute and symmetric norm. This allows us to compute the numerical index of the two-dimensional real L p -space for 3 / 2 ≤ p ≤ 3 .

In this article, we obtain some upper bounds for A-numerical radius of operators in semi-Hilbertian spaces which improve the existing ones. Furthermore, we discuss some generalizations of earlier A-numerical radius inequalities. We also establish several inequalities for B-numerical radius of n × n operator matrices, where B is an n × n diagonal matrix whose diagonal entries are positive operator A

In this paper, the Laplacian matrix of an n-prism network U g n and its applications are studied. Firstly, the relation between the Laplacian matrix of U g n and that of U 0 n is established. Secondly, the analytical expression for the product of all the nonzero Laplacian eigenvalues of U g n is obtained. At the same time, the sum of the reciprocals of all these nonzero Laplacian eigenvalues is deduced

Let R be a prime ring of characteristic different from 2, U the Utumi quotient ring of R, C the extended centroid of R, F and G non-zero generalized derivations of R and f ( x 1 , … , x n ) a multilinear polynomial over C. Suppose that f ( x 1 , … , x n ) is not central valued on R, and R does not embed in M 2 ( K ) , the algebra of 2 × 2 matrices over a field K. We describe all possible forms of F

The minimum rank of a simple graph G is defined to be the smallest possible rank over all symmetric real matrices whose i j t h entry (for i ≠ j ) is nonzero whenever ( i , j ) is an edge in G and is zero otherwise. The sum of the minimum rank of a graph and its maximum nullity (similarly defined) is always the number of vertices in G. This article compares the minimum rank with the clique covering

A characterization of idempotency is given for triangular matrices with entries from an arbitrary ring. This yields an elementary approach to constructing such matrices. It also reveals structure allowing for the explicit enumeration or implicit counting of all idempotent triangular matrices over a given finite ring.

In this paper, we study the probability distribution of eigenvalues of the n×n Euclidean random matrix whose (i,j)th entry is of the form f(∥xi−xj∥2)−unδij∑kf(∥xi−xk∥2) for some real function f. Random points xi's are independently distributed in the lp ellipsoid or on its surface in RN including the unit sphere SN−1, the simplex, the ordinary ellipsoid and the hyper-cube. Here un is allowed to be

We present a characterization of eigenvalue inequalities between two Hermitian matrices by means of inertia indices. As applications, we deal with some classical eigenvalue inequalities for Hermitian matrices, including the Cauchy interlacing theorem and the Weyl inequality, in a simple and unified approach. We also give a common generalization of eigenvalue inequalities for (Hermitian) normalized

Let G be an r-uniform hypergraph of order n. For each p≥1, the p-spectral radius λ(p)(G) is defined as λ(p)(G):=max|x1|p+⋯+|xn|p=1r∑{i1,…,ir}∈E(G)xi1⋯xir. The p-spectral radius was introduced by Keevash-Lenz-Mubayi, and subsequently studied by Nikiforov in 2014. The most extensively studied case is when p = r, and λ(r)(G) is called the spectral radius of G. The α-normal labelling method, which was

We obtain several analogs of real polar decomposition for even dimensional matrices. In particular, we decompose a non-degenerate matrix as a product of a Hamiltonian and an anti-symplectic matrix and under additional requirements we decompose a matrix as a skew-Hamiltonian and a symplectic matrix. We apply our results to study bosonic Gaussian channels up to inhomogeneous symplectic transforms.

Let μ1 be a complex number in the numerical range W(A) of a normal matrix A. In the case when no eigenvalues of A lie in the interior of W(A), we identify the smallest convex region containing all possible complex numbers μ2 for which μ1∗0μ2 is a 2-by-2 compression of A.

Let H be a separable complex Hilbert space with dim H ≥ 3, Bs(H) be the Lie algebra of all bounded self-adjoint operators on H, and let F:iBs(H)→[d,∞] with d≥ 0 be a radial unitary similarity invariant function. In this paper, a structure feature is obtained for maps φ on Bs(H) satisfying F(φ(A)φ(B)−φ(B)φ(A))=F(AB−BA) for all A,B∈Bs(H). As applications, we show that, for a surjective map φ on Bs(H)

In this paper, all connected graphs with three distinct signless Laplacian eigenvalues of which the second one is at most five are determined.

A Lie superalgebra H satisfying H2=Z(H) is called generalized Heisenberg Lie superalgebra. If d(H):=(m∣n) is the minimum number of generators required to describe H, then in this article we intend to find the structure of H when precisely dim⁡(H2)=12[(m+n)2+(n−m)]. Further, we give some results about the capability, and the Schur multiplier of H. Moreover, we find multiplier M(L) for any nilpotent

Let H and K be infinite dimensional separable Hilbert spaces. For A∈LR(H), B∈LR(K) and C∈LR(K,H), we denote by MC=AC0B the upper triangular relation matrix. In this paper, the sets ⋂C∈CM(K,H)σ(MC), ⋂C∈LR(K,H)σ1(MC) and ⋂C∈B(K,H)σ2(MC) are characterized, where σ1∈{σp,σr,σc} and σ2∈{σ,σp,σr,σc}. Moreover, the relationship between σ(MC), σ(A) and σ(B) is described for given relations A∈BCR(H), B∈BCR(K)

A signed graph is a graph whose edges carry the weight ‘+’ or ‘−’. A signed graph S is called sign-regular if d−(v) is same for all v∈V and d+(v) is same for all v∈V. The problems of embedding (i,j)-sign-regular signed graphs in (i+k,j+l)-sign-regular signed graphs is one of the fascinating problem from application point of view which is dealt in this paper with insertion of least number of vertices

Let L be a Lie superalgebra over a field of characteristic different from 2, 3 and write ID∗(L) for the Lie superalgebra consisting of superderivations mapping L to L2 and the central elements to zero. In this paper we first give an upper bound for the superdimension of ID∗(L) by means of linear vector space decompositions. Then we characterize the ID∗-superderivation superalgebras for the nilpotent

We give a complete classification of real and complex Lie superalgebras based on the Heisenberg Lie algebra h2n+1. This classification generalizes previous classifications.

Given disjoint non-empty subsets S and T of {1,…,n−1}, a digraph D with the vertex set {1,2,…,n} is called a directed Toeplitz graph provided the arc i→j occurs if and only if i−j∈S or j−i∈T. We investigate strong connectivity and primitivity of directed Toeplitz graphs. We prove that any primitive directed Toeplitz graph with n≥6 vertices has exponent at least 3 and for each n≥6, there is a primitive

In practice, the time variable cannot be negative. The square integrable function space L2(R+) defined on the half real line R+=(0,∞) models the causal signal space. Given a>1, this paper addresses Fa-frames in the setting of subspaces of L2(R+). We establish the connection between subspace Fa-frames and orthogonal projections; characterize all bounded linear operators on L2(Z×Ta) that transform Fa-frames

In this paper, we investigate the invertibility of generalized g-Bessel multipliers. Sufficient and necessary conditions for invertibility are determined depending on the optimal g-frame bounds. Moreover, we show that, for semi-normalized symbols, the inverse of any invertible generalized g-frame multiplier can be represented as a generalized g-frame multiplier with the reciprocal symbol and dual g-frames

In this paper, we find the roots of lightlike quaternions. By introducing the concept of the Moore-Penrose inverse in split quaternions, we solve the linear equations axb = d, xa = bx, xa=bx¯ and axb = x. As applications, we obtain necessary and sufficient conditions for two split quaternions to be similar or pseudosimilar.

We study a new class NC of Hilbert space operators, named C-normal operators for C a conjugation on a complex Hilbert space H, which were introduced by M. Ptak, K. Simik and A. Wicher. We provide a refined polar decomposition of C-normal operators. It is proved that those invertible ones are norm dense in NC and each contraction in NC is a mean of two unitary ones. For a dense class of operators, we

Frobenius' Theorem states that, besides the fields of real and complex numbers, the algebra of quaternions H is the only finite-dimensional real division algebra. We first give a short elementary proof of this theorem, then characterize finite-dimensional real algebras that contain either a copy of C, a copy of H, or a pair of anticommuting invertible elements through the dimensions of their (left)

It is well-known that the lattice of all submodules of a module is modular. However, this is not the case for the lattice of subsemimodules of a semimodule. We show examples and describe these lattices for a given semimodule. We study closed and splitting subsemimodules and submodules of a given semimodule or module M, respectively. We derive a sufficient condition under which the lattice Lc(M) of

A complex unit gain graph (or T-gain graph) is a triple Φ=(G,T,ϕ) (or (G,ϕ) for short) consisting of a simple graph G with |V(G)|=n, as the underlying graph of (G,ϕ), the set of unit complex numbers T={z∈C:|z|=1} and a gain function ϕ:E→→T with the property that ϕ(eij)=ϕ(eji)−1=ϕ(eji)¯. The adjacency matrix of (G,ϕ) is A(G,ϕ)=(aij)n×n, where aij=ϕ(eij) if vi is adjacent to vj and aij=0 otherwise. The

In this paper, we investigate structured backward errors for two kinds of generalized saddle point systems. The explicit formulas of structured backward errors are derived, which are useful for testing the stability of practical algorithms.

In this work, we present and prove the explicit formula for the determinant of a class of n×n nonsymmetric Toeplitz matrices. Setting up one of the nonzero subdiagonals to zero results in special pentadiagonal Toeplitz matrices, whose determinant formulas are conjectured by Anđelić and Fonseca in Anđelić [Some determinantal considerations for pentadiagonal matrices. Linear Multilinear Algebra. 2020

Bhatia, Lim, and Yamazaki studied the norm minimality of several Kubo-Ando means of positive semidefinite matrices. Recently, Hiai proved a norm minimality result involving the the weighted geometric mean A♯1−αB=A1/2(A−1/2BA−1/2)1−αA1/2, 0≤α≤1 and its ‘naïve’ extension given by Qα,z(A,B)=(B1−α2zAαzB1−α2z)z, which is a matrix function in the definition of the quantum α-z-Rényi divergence. In connection

In this paper, we consider a linear constraint problem of Hermitian unitary symplectic matrices and its approximation. By constructing a simple unitary matrix U, we verify that Hermitian unitary symplectic matrices are unitary similar to block diagonal Hermitian unitary matrices via U, which simplifies and is crucial to solving the linear constraint problem, and is a special feature of this paper.

The main topic of this paper is the group invertibility and expressions of group inverses of linear operators in Banach spaces. We first give some properties of regular factorizations and then derive some new characterizations of the group invertibility of linear operators. Based on these results, we use any one of generalized inverses to give some concise expressions of the group inverse.

In this paper we describe central extensions (up to isomorphism) of all complex null-filiform and filiform Zinbiel algebras. It is proven that every non-split central extension of an n-dimensional null-filiform Zinbiel algebra is isomorphic to an (n+1)-dimensional null-filiform Zinbiel algebra. Moreover, we obtain all pairwise non isomorphic quasi-filiform Zinbiel algebras.

Two numerical algorithms are proposed for computing an interval matrix containing the matrix gamma function. In 2014, the author presented algorithms for enclosing all the eigenvalues and basis of invariant subspaces of A∈Cn×n. As byproducts of these algorithms, we can obtain interval matrices containing diagonal blocks whose spectrums are included in that of A. In this paper, we interpret the interval

This paper is concerned with stability of deficiency indices of Hermitian subspaces (i.e. linear relations) under relatively bounded perturbations in Hilbert spaces. Several results about invariance of deficiency indices of Hermitian subspaces under relatively bounded perturbations are established. As a consequence, invariance of self-adjointness of Hermitian subspaces under relatively bounded perturbations

For a family of real matrices with entries polynomially depending on parameters, symbolic algorithms are proposed for verification of the Routh – Hurwitz stability under parameter variations in a given box, and for the distance to instability computation in the case of perturbations over C. Both problems are reduced to the analysis of real zeros of a pair of univariate polynomials; one of these reductions

We investigate the number of zero entries in a unitary matrix. We show that the sets of numbers of zero entries for n×n unitary and orthogonal matrices are the same. They are both the set {0,1,…,n2−n−4,n2−n−2,n2−n} for n>4. We explicitly construct examples of orthogonal matrices with the numbers in the set. We apply our results to construct a necessary condition by which a multipartite unitary operation

We study the one-sided version of central Drazin inverses. An element a in a ring R is said to be left central Drazin invertible if there is x∈R such that xa∈C(R), xan+1=an for some n∈N. The right central Drazin invertible elements can be defined similarly. It is shown that an element a∈R is central Drazin invertible if and only if a is both left and right central Drazin invertible. Some well-known

The quadratic numerical range has an important application to the spectral distribution of operators. In this paper, the geometry of the quadratic numerical range is investigated. Moreover, some sufficient conditions for the quadratic numerical range to consist of two disjoint components are discussed. Finally, the necessary and sufficient condition for the quadratic numerical range to be contained

In this paper, we focus on the dihedral groups and the dicyclic groups, and consider their corresponding integral Cayley graphs. We obtain the sufficient conditions for the integrality of the distance powers ΓD of the Cayley graph Γ=X(D2n,S) (resp. Γ=X(T4n,S)) ( n≥3) for a set of nonnegative integers D. In particular, for a prime p, we show that if Γ=X(D2p,S) (resp. Γ=X(T4p,S)) is integral, then the

Let ℜn,β be the family of graphs with n vertices and with the matching number β. The distance signless Laplacian spectral radius of a graph G is denoted by ρ(G). Li and Sun (Matching number, connectivity and eigenvalues of distance signless Laplacians. Linear Multilinear Algebra, 2019. https://doi.org/10.1080/03081087.2019.1588218) proved that if n≥72β, ρ(G)≥ρ(Kβ⋁Kn−β¯) for any G∈ℜn,β, with equality

In this paper, we establish some numerical radius inequalities for 2×2 bounded linear operator defined on a complex Hilbert space. As a natural application, the existence of the all polynomial zeros is identified in a specific small disk. Moreover, we provide a refinement of an earlier numerical radius inequality due to Herzallah et al. [Numerical radius inequalities for certain 2×2 operator matrices

In this paper, we prove that every local derivation on Witt algebras Wn,Wn+ or Wn++ is a derivation for any n∈N. As a consequence we obtain that every local derivation on a centerless generalized Virasoro algebra of higher rank is a derivation.

A λm-connected graph G is optimally m-restricted edge-connected if its m-restricted edge-connectivity λm(G) equals the minimum m-edge-degree ξm(G), and super m-restricted edge-connected if all minimum m-restricted edge cuts are trivial. In this study, a bound for the size of graphs with girth greater than any given integer l (l≥3) is obtained firstly. Then the second minimum degree conditions for optimally

We determine that the maximum rank of an order-n (≥2) tensor with format 2×⋯×2 over the finite field F2 is 2⋅3n/2−1 for even n, and 3⌊n/2⌋ for odd n. Since tensor rank is non-increasing upon taking field extensions, F2 gives the largest rank attainable for this tensor format. We also determine a maximum rank canonical form and compute its orbit under the action of the symmetry group GL2(F2)×n, and

In this paper we give two characterizations of those involutions on the algebra of quaternions that are automorphisms or anti-automorphisms, one in terms of inner automorphisms and the other in terms of the involution eigenspaces.

Let T be a triangular n-matrix ring ( n≥ 2). It is shown that, under some mild assumptions, a map δ:T→T is a multiplicative Lie derivation if and only if δ(X)=d(X)+γ(X) holds for all X∈T, where d:T→T is an additive derivation and γ:T→Z(T) is a central-valued map vanishing on all commutators.

We introduce a new operator – a modified λ-differential operator – from a usual differential operator. We construct the free (modified) λ-differential (k,∂)-algebra on a (modified) differential (k,∂)-module, and apply it to erect the universal enveloping (modified) differential algebra of a (modified) λ-differential Lie (k,∂)-algebra. The corresponding Poincaré-Birchoff-Witt theorems are obtained.

In this paper, we consider the symmetric and Hankel-symmetric transportation polytope Ut&h(R,S), which is the convex set of all symmetric and Hankel-symmetric non-negative matrices with prescribed row sum vector R and prescribed column sum vector S. We characterize all extreme points of Ut&h(R,S). Moreover, we show that the extreme points of Ωnt&h, the polytope of symmetric and Hankel-symmetric doubly

In this paper, by using the special structure of the real representation of quaternion matrices, the properties of the Moore–Penrose inverse of quaternion matrices and the Kronecker product of matrices, we study least-squares bihermitian and skew bihermitian solutions of the quaternion matrix equation AXB = C, respectively. First we study the special structures of quaternion bihermitian and skew bihermitian

We completely characterize the left-symmetric points, the right-symmetric points, and the symmetric points in the sense of Birkhoff-James, in a Banach space. We obtain a complete characterization of the left-symmetric (right-symmetric) points in the infinity sum of two Banach spaces, in terms of the left-symmetric (right-symmetric) points of the constituent spaces. As an application of this characterization

In this article, we present the restricted numerical for the Laplacian matrix of a directed graph (digraph). We motivate our interest in the restricted numerical range by its close connection to the algebraic connectivity of a digraph. Moreover, we show that the restricted numerical range can be used to characterize digraphs, some of which are not determined by their Laplacian spectrum. Finally, we

In this paper, we obtain new additive perturbation bounds of the Moore-Penrose inverse under the unitarily invariant norm and the Q-norm by using the singular value decomposition, respectively. These bounds always improve the corresponding ones in [Cai L et al. Additive and multiplicative perturbation bounds for the Moore-Penrose inverse. Linear Algebra Appl. 2011;434:480–489].

(2020). Correction. Linear and Multilinear Algebra. Ahead of Print.

We classify all Rota–Baxter operators of nonzero weight on the matrix algebra of order three over an algebraically closed field of characteristic zero which do not arise from the decompositions of the entire algebra into a direct vector space sum of two subalgebras.

In this article, first we will present the new rigorous perturbation bounds with normwise perturbation for the generalized Cholesky block downdating problem by combining the unified matrix–vector equation approach with the method of Lyapunov majorant functions and the Banach fixed point theorem. Then, we will derive the explicit expressions for the six distinct kinds of condition numbers, i.e. four

The purpose of this paper is to analyse the Harnack part of some truncated shifts whose ρ-numerical radius equal one in the finite dimensional case. As pointed out in Theorem 2.17 of Cassier G, Benharrat M, Belmouhoub S [Harnack parts of ρ-contractions. J Oper Theory. 2018;80(2):453–480], a key point is to describe the null spaces of the ρ-operatorial kernel of these truncated shifts. We establish

Eigenvalue behaviours of Schrödinger operator defined on n-dimensional lattice with n + 1 delta potentials are studied. It can be shown that lower threshold eigenvalue and lower threshold resonance appear for n≥2, and lower super-threshold resonance appears for n = 1.

We consider the class Asym0(n,k) of symmetric (0,1)-matrices with zero trace and constant row sums k which can be identified with the class of the adjacency matrices of k-regular undirected graphs. In a previous paper, two partial orders, the Bruhat and the Bruhat-graph order, have been introduced in this class. In fact, when k = 1 or k = 2, it was shown that the two orders coincide, while for k≥3

Characterizing which graphs have Laplaican perfect state transfer or Laplacian pretty good state transfer is an interesting and desirable problem. In this paper, we investigate the existence of Laplacian perfect state transfer and Laplacian pretty good state transfer in edge coronas. We give sufficient conditions for edge coronas to not have Laplacian perfect state transfer as well as sufficient conditions