Showing papers in "Linear & Multilinear Algebra in 2010"
TL;DR: In this article, the notion of the core inverse is introduced as an alternative to the group inverse and several properties of its properties are derived with a perspective towards possible applications, such as matrix partial ordering.
Abstract: This article introduces the notion of the Core inverse as an alternative to the group inverse. Several of its properties are derived with a perspective towards possible applications. Furthermore, a matrix partial ordering based on the Core inverse is introduced and extensively investigated.
297 citations
TL;DR: Some inequalities for continuous synchronous (asynchronous) functions of selfadjoint linear operators in Hilbert spaces, under suitable assumptions for the involved operators, are given in this article, where the authors consider the case where the operator is a self-adjoint operator.
Abstract: Some inequalities for continuous synchronous (asynchronous) functions of selfadjoint linear operators in Hilbert spaces, under suitable assumptions for the involved operators, are given.
76 citations
TL;DR: In this article, the star order on the algebra L(ℋ) of bounded operators on a Hilbert space ℋ has been studied and a new interpretation of this order has been presented which allows to generalize to this setting many known results for matrices: functional calculus, semi-lattice properties, shorted operators and orthogonal decompositions.
Abstract: We study the star order on the algebra L(ℋ) of bounded operators on a Hilbert space ℋ. We present a new interpretation of this order which allows to generalize to this setting many known results for matrices: functional calculus, semi-lattice properties, shorted operators and orthogonal decompositions. We also show several properties for general Hilbert spaces regarding the star order and its relationship with the functional calculus and the polar decomposition, which were unknown even in the finite-dimensional setting. We also study the existence of strong limits of star-monotone sequences and nets.
39 citations
TL;DR: In this article, two distinct bipartite graphs which are cospectral for both the adjacency and normalized Laplacian matrices were constructed by unfolding a base bipartitite graph in two different ways.
Abstract: In this note we show how to construct two distinct bipartite graphs which are cospectral for both the adjacency and normalized Laplacian matrices by ‘unfolding’ a base bipartite graph in two different ways.
38 citations
TL;DR: In this article, the authors studied some common properties between Lie algebras and their tensor products, and presented some bounds on the nilpotency class and solvability length of L ⊗ K, provided such information is given on L or K.
Abstract: Ellis (G. Ellis, A non-abelian tensor product of Lie algebras, Glasgow Math. J. 39 (1991), pp. 101–120.) introduced the notion of the non-abelian tensor product L ⊗ K for a pair of Lie algebras L, K and investigated some of its fundamental properties. In this article, we study some common properties between Lie algebras and their tensor products, and present some bounds on the nilpotency class and solvability length of L ⊗ K, provided such information is given on L or K. Also, we give some upper and lower bounds for the dimension of L ⊗ K if L and K are finite-dimensional nilpotent Lie algebras and ideals of a single Lie algebra.
33 citations
TL;DR: In this paper, the Moore-Penrose invertibility of rings with a general involution was studied and necessary and sufficient conditions for aa = bb were given for a general ring with an arbitrary involution.
Abstract: In this article, we consider Moore–Penrose invertibility in rings with a general involution. Given two von Neumann regular elements a, b in a general ring with an arbitrary involution, we aim to give necessary and sufficient conditions to aa † = bb †. As a special case, EP elements are considered.
29 citations
TL;DR: In this paper, it was shown that the Lie algebra of skew-symmetric matrices with respect to either transpose or symplectic involution is zero product determined, and that every bilinear map of a vector space X is of the form X = T ([x, y]) for some linear map T provided that the following condition is fulfilled.
Abstract: We show that the Lie algebra ℒ of skew-symmetric matrices with respect to either transpose or symplectic involution is zero product determined. This means that every bilinear map {·,·} from ℒ × ℒ into a vector space X is of the form {x, y} = T ([x, y]) for some linear map T provided that the following condition is fulfilled: [x, y] = 0 implies {x, y} = 0.
26 citations
TL;DR: In this article, the authors characterized all regular graphs whose second largest eigenvalue does not exceed 1 and determined all coronas, different from cones, with the same property, and some results and examples regarding unsolved cases are also given.
Abstract: We characterize all regular graphs whose second largest eigenvalue does not exceed 1. In the sequel, we determine all coronas, different from cones, with the same property. Some results and examples regarding unsolved cases are also given.
23 citations
TL;DR: In this article, necessary and sufficient conditions for 2-by-2 block operator valued triangular matrices to be Moore-Penrose (MP) invertible were obtained and new representations of such MP inverses were given in terms of the individual blocks.
Abstract: We obtain necessary and sufficient conditions for 2-by-2 block operator valued triangular matrices to be Moore–Penrose (MP) invertible and give new representations of such MP inverses in terms of the individual blocks.
23 citations
TL;DR: In this paper, necessary and sufficient conditions are given for the existence of the group inverses of block matrix, where and exist, A = c 1 B + c 2 C, non-zero elements c 1 and c 2 are in the centre of K and block matrix, where A = B k C l, k and l are positive integers.
Abstract: Let K be any skew field and K m×n be the set of all the m × n matrices over K. In this article, necessary and sufficient conditions are given for the existence of the group inverses of block matrix , where and exist, A = c 1 B + c 2 C, non-zero elements c 1 and c 2 are in the centre of K and block matrix , where A = B k C l , k and l are positive integers. Then the representations of the group inverses of these block matrices are also given.
21 citations
TL;DR: Tan et al. as mentioned in this paper proved divisibility among power GCD matrices and among power LCM matrices on two coprime divisor chains, where each element of S 1 is co-partitioned with S 2, where S 1 and S 2 are divisors.
Abstract: Let a, b and h be positive integers and S = {x 1, … , x h } be a set of h distinct positive integers. The set S is called a divisor chain if there is a permutation σ of {1, … , h} such that x σ(1)|…|x σ(h). We say that the set S consists of two coprime divisor chains if we can partition S as S = S 1 ∪ S 2, where S 1 and S 2 are divisor chains and each element of S 1 is coprime to each element of S 2. The matrix having the ath power (x i , x j ) a of the greatest common divisor (GCD) of x i and x j as its (i,j)-entry is called the ath power GCD matrix defined on S, denoted by (S a ). Similarly, we can define the ath power least common multiple (LCM) matrix [S a ]. In the first paper of the series, Tan [Q. Tan, Divisibility among power GCD matrices and among power LCM matrices on two coprime divisor chains, Linear Multilinear Algebra 58 (2010), pp. 659--671] showed that if S consists of two coprime divisor chains and 1 ∈ S and a|b, then (S a )|(S b ), [S a ]|[S b ] and (S a )|[S b ] hold in the ring M h (Z)...
TL;DR: In this paper, the reverse order laws for {1, 2, 3, 4, 5, 6, 7}-inverses of multiple matrix products by using the maximal and minimal ranks of the generalized Schur complements were studied.
Abstract: In this article, we study the reverse order laws for {1, 2, 3}- and {1, 2, 4}-inverses of multiple matrix products by using the maximal and minimal ranks of the generalized Schur complements. The necessary and sufficient conditions for the inclusions and are presented.
TL;DR: In this article, the range space, null space, nonsingularity and group invertibility of linear combinations T = c 1 T 1 + c 2 T 2 of two k-potent matrices T 1 and T 2 were studied.
Abstract: An n × n complex matrix A is said to be k-potent if A k = A. Let T 1 and T 2 be k-potent and c 1 and c 2 be two nonzero complex numbers. We study the range space, null space, nonsingularity and group invertibility of linear combinations T = c 1 T 1 + c 2 T 2 of two k-potent matrices T 1 and T 2.
TL;DR: In this article, a generalization of a known result about the subdirect sum of two S-SDD matrices is obtained for Σ-SDDs matrices, which is a subclass of H-matrices.
Abstract: In this article, a generalization of a known result about the subdirect sum of two S-SDD (strictly diagonally dominant) matrices is obtained for Σ-SDD matrices. The class of Σ-SDD matrices is a generalization of S-SDD matrices, and it is also a subclass of H-matrices. More precisely, the question of when the subdirect sum and, consequently, the usual sum of two Σ-SDD matrices is an Σ-SDD matrix is studied.
TL;DR: In this paper, the authors investigated some properties concerning lineability and spaceability of the (p, q)-summing set of a continuous n-homogeneous polynomial P : E ⟹ F between Banach spaces and 1 ≤ q ≤ p < ∞.
Abstract: Given a continuous n-homogeneous polynomial P : E ⟹ F between Banach spaces and 1 ≤ q ≤ p < ∞, in this article we investigate some properties concerning lineability and spaceability of the (p; q)-summing set of P, defined by S p; q (P) = {a ∈ E : P is (p; q)-summing at a}.
TL;DR: In this paper, the authors show how to construct infinite families of Laplacian integral graphs using Kronecker products, balanced incomplete block designs, and solutions to certain Diophantine equations.
Abstract: The aim of this article is to answer a question posed by Merris in European Journal of Combinatorics, 24 (2003) pp. 413 − 430, about the possibility of finding split non-threshold graphs that are Laplacian integral, i.e. graphs for which the eigenvalues of the corresponding Laplacian matrix are integers. Using Kronecker products, balanced incomplete block designs, and solutions to certain Diophantine equations, we show how to build infinite families of these graphs.
TL;DR: In this article, a new operator equality in the framework of Hilbert C*-modules is presented, which is an extension of the Euler-Lagrange type identity in the setting of Hilbert bundles as well as several generalized operator Bohr's inequalities due to O. Hirzallah, W.-S. Cheung, J.E. Pecaric and F. Zhang.
Abstract: We present a new operator equality in the framework of Hilbert C*-modules. As a consequence, we get an extension of the Euler–Lagrange type identity in the setting of Hilbert bundles as well as several generalized operator Bohr's inequalities due to O. Hirzallah, W.-S. Cheung, J.E. Pecaric and F. Zhang.
TL;DR: In this paper, it was shown that the unit I is a Jordan full-derivable point of prime Banach algebras containing a non-trivial idempotent, factor von Neumann (FVNE) algebra.
Abstract: Let be a unital ring. An element is said to be a Jordan full-derivable point of if every additive map δ from into itself Jordan derivable at Z (i.e. δ(A)B + Aδ(B) + δ(B)A + Bδ(A) = δ(Z) for every A, B ∈ with AB + BA = Z) is a Jordan derivation. In this article, under some mild conditions on unital prime ring or triangular ring , it is shown that the unit I is a Jordan full-derivable point of . Particularly, the unit I is a Jordan full-derivable point of prime Banach algebras containing a non-trivial idempotent, factor von Neumann algebras, and nest algebras.
TL;DR: In this paper, the first and second-minimal permanents of the Laplacian matrix of trees and bipartite graphs were derived for G in ℱ n,Δ and characterized the extremal graphs.
Abstract: Let G be a graph and L(G) be the Laplacian matrix of G. When G is a tree or a bipartite graph, Brualdi and Goldwasser [Permanent of the Laplacian matrix of trees and bipartite graphs, Discrete Math., 48 (1984), pp. 1–21] characterized the bounds for the permanent of L(G). In this article, as the continuance of it, we study the lower bounds for the permanent of L(G) when G is a tree. Here are some of our results: (1) Let 𝒯 n,k be the collection of trees on n vertices with at most k pendent vertices and let ≔ {G ∈ 𝒯 n,k : there is a pendent vertex, say v, in G such that G − v ∈ 𝒯 n−1,k−1}. We determine the minimum permanent of L(G) for G in and characterize the extremal graph. (2) Let ℱ n,Δ be the set of trees on n vertices with maximum degree Δ. We determine the first- and the second-minimal permanents of L(G) for G in ℱ n,Δ and characterize the extremal graphs. (3) Let 𝒢 n,m be the collection of n-vertex trees each of which has an m-matching, where m ≥ 3. We determine the second- and the third-minimal per...
TL;DR: Boyle and Handelman as mentioned in this paper characterized all lists of n complex numbers that can be the nonzero spectrum of a nonnegative matrix and presented a constructive proof for this result in the special case when the lists are real and contain two positive numbers and n − 2 negative numbers.
Abstract: Boyle and Handelman [M. Boyle and D. Handelman, The spectra of nonnegative matrices via symbolic dynamics, Ann. Math. 133 (1991), pp. 249–316.] characterized all lists of n complex numbers that can be the nonzero spectrum of a nonnegative matrix. This article presents a constructive proof of this result in the special case when the lists are real and contain two positive numbers and n − 2 negative numbers. A bound for the number of zeros that needs to be added to the list to achieve a nonnegative realization is presented in this case.
TL;DR: In this article, the spectral radius of linear combinations of two projections in C*-algebras was studied. And the commutator of the two projections was investigated.
Abstract: In this note, we study the spectrum and give estimations for the spectral radius of linear combinations of two projections in C*-algebras. We also study the commutator of two projections.
TL;DR: In this article, the authors give a thorough discussion of additive maps between nest algebras acting on Banach spaces which preserve rank-one operators in both directions, and show that additive maps can be used to preserve rank one operators in two directions.
Abstract: In this article, we give a thorough discussion of additive maps between nest algebras acting on Banach spaces which preserve rank-one operators in both directions.
TL;DR: In this article, the authors consider the problem of finding natural representations of the groups G U(2n, /0) and H = G SL(V) SU(2 n, / 0) on the k-th exterior power of a 2n-dimensional vector space over a field equipped with a Hermitian form f of Witt index n epsilon 1.
Abstract: Let V be 2n-dimensional vector space over a field equipped with a nondegenerate skew--Hermitian form f of Witt index n epsilon 1, let 0 be the fix field of and let G denote the group of isometries of (V, f). For every k {1, ..., 2n}, there exist natural representations of the groups G U(2n, /0) and H = G SL(V) SU(2n, /0) on the k-th exterior power of V. With the aid of linear algebra, we prove some properties of these representations. We also discuss some applications to projective embeddings and hyperplanes of Hermitian dual polar spaces.
TL;DR: In this article, it was shown that a derivation from into to a strongly double triangle subspace lattice is quasi-spatial and local derivation is local derivative.
Abstract: Let be a strongly double triangle subspace lattice. It is proved that a derivation δ from into is quasi-spatial. It is also shown that if Δ is derivable at zero, i.e. if Δ(A)B + AΔ(B) = 0 for all A and B in with AB = 0, then Δ(A) = δ(A) + λA for all where δ is a derivation and λ is a scalar. It is also shown that a local derivation from into is a derivation.
TL;DR: In this article, it was shown that for each permutation w containing no decreasing subsequence of length k, the Kazhdan-Lusztig immanant Imm w (x) vanishes on all matrices having k equal rows or columns.
Abstract: We show that for each permutation w containing no decreasing subsequence of length k, the Kazhdan–Lusztig immanant Imm w (x) vanishes on all matrices having k equal rows or columns. Also, we define two filtrations of the vector space of immanants via products of matrix minors and pattern avoidance and use the above result to show that these filtrations are equivalent. Finally, we construct new and simple inequalities satisfied by the minors of totally nonnegative matrices.
TL;DR: In this paper, an inclusion region for the eigenvalues of a matrix that can be considered an alternative to the Brauer set is derived, accompanied by non-singularity conditions.
Abstract: We derive an inclusion region for the eigenvalues of a matrix that can be considered an alternative to the Brauer set. It is accompanied by non-singularity conditions.
TL;DR: In this article, the authors determine graphs with the largest spectral radius among all the cacti with n vertices and k-pendant vertices, and show that these graphs are the ones with the smallest spectral radius.
Abstract: A graph G is a cactus if any two of its cycles have at most one common vertex. In this article, we determine graphs with the largest spectral radius among all the cacti with n vertices and k-pendant vertices.
TL;DR: In this article, the largest Laplacian eigenvalue λ 1(G) of a simple graph with n vertices, m edges, diameter D and degree sequence d 1, d 2, d 3, d 4, d 5, d 6, d 7, d 8, d 9, d 10, d 11, d 12, d 14, d 15, d 16, d 17, d 18, d 19, d 20, d 21, d 22, d 23, d 24,
Abstract: Let G be a simple graph with n vertices, m edges, diameter D and degree sequence d 1, d 2, …, d n , and let λ1(G) be the largest Laplacian eigenvalue of G. Denote Δ = max{d i : 1 ≤ i ≤ n}, and , where α is a real number. In this article, we first give an upper bound on λ1(G) for a non-regular graph involving Δ and D; next present two upper bounds on λ1(G) for a connected graph in terms of d i and (α m) i ; at last obtain a lower bound on λ1(G) for a connected bipartite graph in terms of d i and (α t) i . Some known results are shown to be the consequences of our theorems.
TL;DR: In this article, the authors examined the greatest common unitary divisor (GCUD) reciprocal LCUM matrices and derived the determinant of these matrices with respect to certain types of functions arising from the LCUM problematics.
Abstract: A divisor d ∈ ℤ+ of n ∈ ℤ+ is said to be a unitary divisor of n if (d, n/d) = 1. In this article we examine the greatest common unitary divisor (GCUD) reciprocal least common unitary multiple (LCUM) matrices. At first we concentrate on the difficulty of the non-existence of the LCUM and we present three different ways to overcome this difficulty. After that we calculate the determinant of the three GCUD reciprocal LCUM matrices with respect to certain types of functions arising from the LCUM problematics. We also analyse these classes of functions, which may be referred to as unitary analogs of the class of semimultiplicative functions, and find their connections to rational arithmetical functions. Our study shows that it does make a difference how to extend the concept of LCUM.
TL;DR: In this article, a general form of a surjective (not assumed additive) mapping φ, preserving the nonzero idempotency of a certain product, was given, where φ is additive and either multiplicative or antimultiplicative.
Abstract: We obtain a general form of a surjective (not assumed additive) mapping φ, preserving the nonzero idempotency of a certain product, being defined (a) on the algebra of all bounded linear operators B(X), where X is at least three-dimensional real or complex Banach space, (b) on the set of all rank-one idempotents in B(X) and (c) on the set of all idempotents in B(X). In any of the cases it turns out that φ is additive and either multiplicative or antimultiplicative.