# Showing papers in "Linear & Multilinear Algebra in 2015"

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TL;DR: In this paper, the uniqueness and construction of the Z matrix in Theorem 2.1 was shown and an affirmative answer to a question proposed in [J. Math. Anal. Appl. 407 (2013) 436-442 was given.

Abstract: We show the uniqueness and construction (of the Z matrix in Theorem 2.1, to be exact) of a matrix decomposition and give an affirmative answer to a question proposed in [J. Math. Anal. Appl. 407 (2013) 436-442].

69 citations

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TL;DR: In this paper, the spectral radius of a graph with minimum degree is shown to be the largest eigenvalue of the adjacency matrix of the graph, and two previous theorems due to Fiedler and Nikiforov and Lu et al. are obtained.

Abstract: Let be a graph with minimum degree . The spectral radius of , denoted by , is the largest eigenvalue of the adjacency matrix of . In this note, we mainly prove the following two results.(1) Let be a graph on vertices with . If , then contains a Hamilton path unless .(2) Let be a graph on vertices with . If , then contains a Hamilton cycle unless . As corollaries of our first result, two previous theorems due to Fiedler and Nikiforov and Lu et al. are obtained, respectively. Our second result refines another previous theorem of Fiedler and Nikiforov.

56 citations

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TL;DR: In this paper, it was proved that a symmetric tensor is copositive if and only if each of its principal sub-tensors has no (non-positive) negative -eigenvalue.

Abstract: In this paper, it is proved that a symmetric tensor is (strictly) copositive if and only if each of its principal sub-tensors has no (non-positive) negative -eigenvalue. Necessary and sufficient conditions for (strict) copositivity of a symmetric tensor are also given in terms of -eigenvalues of the principal sub-tensors of that tensor. This presents a method for testing (strict) copositivity of a symmetric tensor by means of lower dimensional tensors. Also, an equivalent definition of strictly copositive tensors is given on the entire space .

53 citations

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TL;DR: In this paper, the authors studied the spectral properties of the connected odd-bipartite hypergraphs and showed that the Laplacian H-spectrum and signless H-Spectrum of a connected -uniform hypergraph are equal if and only if the hypergraph is even and is odd.

Abstract: A -uniform hypergraph is called odd-bipartite, if is even and there exists some proper subset of such that each edge of contains odd number of vertices in . Odd-bipartite hypergraphs are generalizations of the ordinary bipartite graphs. We study the spectral properties of the connected odd-bipartite hypergraphs. We prove that the Laplacian H-spectrum and signless Laplacian H-spectrum of a connected -uniform hypergraph are equal if and only if is even and is odd-bipartite. We further give several spectral characterizations of the connected odd-bipartite hypergraphs. We also give a characterization for a connected -uniform hypergraph whose Laplacian spectral radius and signless Laplacian spectral radius are equal; thus, provide an answer to a question raised by L. Qi. By showing that the Cartesian product of two odd-bipartite -uniform hypergraphs is still odd-bipartite, we determine that the Laplacian spectral radius of is the sum of the Laplacian spectral radii of and , when and are both connected odd-bipa...

51 citations

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TL;DR: In this paper, an application of resistance distances to the bipartiteness of graphs is given, and an interlacing inequality for eigenvalues of the resistance matrix and the Laplacian matrix is given.

Abstract: In this paper, we obtain formulas for resistance distances and Kirchhoff index of subdivision graphs. An application of resistance distances to the bipartiteness of graphs is given. We also give an interlacing inequality for eigenvalues of the resistance matrix and the Laplacian matrix.

51 citations

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TL;DR: In this paper, the core inverse of a linear multilinear algebra is shown to be the unique solution of and, and several characterizations of core inverse, the core partial ordering and the reverse order law for core inverse are established.

Abstract: In this note, we revisit the core inverse and the core partial ordering introduced by Baksalary and Trenkler [Linear Multilinear Algebra. 2010;58:681–697]. We prove that the core inverse of is the unique solution of and , and establish several characterizations of the core inverse, the core partial ordering and the reverse order law for the core inverse.

50 citations

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TL;DR: In this paper, it was shown that every Lie triple derivation on is of the form, where is a derivation of, is a singular Jordan derivation, and is a linear mapping from to its centre that vanishes on.

Abstract: Let be a unital algebra with a nontrivial idempotent over a unital commutative ring . We show that under suitable assumptions, every Lie triple derivation on is of the form , where is a derivation of , is a singular Jordan derivation of and is a linear mapping from to its centre that vanishes on . As an application, we characterize Lie triple derivations and Lie derivations on triangular algebras and on matrix algebras.

39 citations

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[...]

Jilin University

^{1}TL;DR: The notion of omni-Hom-Lie algebras was introduced in this paper, which is an algebra associated to a vector space and an invertible linear map.

Abstract: In this paper, first we show that is a Hom–Lie algebra if and only if is an differential graded-commutative algebra. Then, we revisit representations of Hom–Lie algebras and show that there are a series of coboundary operators. We also introduce the notion of an omni-Hom–Lie algebra associated to a vector space and an invertible linear map. We show that regular Hom–Lie algebra structures on a vector space can be characterized by Dirac structures in the corresponding omni-Hom–Lie algebra. The underlying algebraic structure of the omni-Hom–Lie algebra is a Hom–Leibniz algebra, or a Hom–Lie 2-algebra.

37 citations

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TL;DR: In this paper, a necessary and sufficient condition for the existence of a solution to the classical system of matrix equations by using generalized inverses is presented, and the general solution is given when it is solvable.

Abstract: In this paper, we consider an expression of the general solution to the classical system of matrix equationsWe present a necessary and sufficient condition for the existence of a solution to the system by using generalized inverses. We give an expression of the general solution to the system when it is solvable. As applications, we derive some necessary and sufficient conditions for the consistence to the systemand the systemwhere means conjugate transpose. We also give the expressions of the general solutions to the systems.

36 citations

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TL;DR: In this paper, the principal powers of complex square matrices with positive definite real part were studied and the notion of geometric mean was extended to such matrices and an operator norm bound was established.

Abstract: We study principal powers of complex square matrices with positive definite real part, or with numerical range contained in a sector. We extend the notion of geometric mean to such matrices and we establish an operator norm bound in this context.

36 citations

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TL;DR: In this article, the Young-type inequalities for positive real numbers and their matrix analogues for positive definite matrices were presented and compared to the matrix analogue of the double-inequality.

Abstract: We present several new Young-type inequalities for positive real numbers and we apply our results to obtain the matrix analogues. Among others, for real numbers , and , with and , we prove the inequalitieswhere and are, respectively, the (weighted) arithmetic and geometric means of the positive real numbers and with . In addition, we show that both bounds are sharp. An example of a matrix analogue for the case is the double-inequalityfor positive definite matrices . Our results extend some fresh inequalities established by Kittaneh, Manasrah, Hirzallah and Feng. Estimates for the quotient and its matrix analogues given by Furuichi and Minculete are also improved.

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TL;DR: In this article, the authors give graph theoretical formulas for the trace of a tensor which do not involve the differential operators and auxiliary matrix and give a characterization (in terms of the traces of the adjacency tensors) of uniform hypergraphs whose spectra are -symmetric.

Abstract: We give some graph theoretical formulas for the trace of a tensor which do not involve the differential operators and auxiliary matrix. As applications of these trace formulas in the study of the spectra of uniform hypergraphs, we give a characterization (in terms of the traces of the adjacency tensors) of the -uniform hypergraphs whose spectra are -symmetric, thus give an answer to a question raised in Cooper and Dutle [Linear Algebra Appl. 2012;436:3268–3292]. We generalize the results in Cooper and Dutle [Linear Algebra Appl. 2012;436:3268–3292, Theorem 4.2] and Hu and Qi [Discrete Appl. Math. 2014;169:140–151, Proposition 3.1] about the -symmetry of the spectrum of a -uniform hypergraph, and answer a question in Hu and Qi [Discrete Appl. Math. 2014;169:140–151] about the relation between the Laplacian and signless Laplacian spectra of a -uniform hypergraph when is odd. We also give a simplified proof of an expression for and discuss the expression for .

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TL;DR: In this paper, a generalization of the operator inequalities due to Lin [J. Math. Anal. Appl. 2013;402:127-132] and [Studia Math. Appl., 2013;215:187-194] is presented.

Abstract: In this note, we generalize some operator inequalities due to Lin [J. Math. Anal. Appl. 2013;402:127–132] and [Studia Math. 2013;215:187–194] as follows: Let and be positive operators on a Hilbert space with Then for and every positive unital linear map

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TL;DR: In this article, the adjacency spectra of four types of graph operations on and involving the R-graph of a regular graph and an arbitrary graph were determined for each edge of the regular graph by adding a new vertex for each vertex and joining each new vertex to the end vertices of the corresponding edge.

Abstract: For a regular graph and an arbitrary graph , we determine the adjacency (respectively, Laplacian and signless Laplacian) spectra of four types of graph operations on and involving the R-graph of , obtained from by adding a new vertex for each edge of and joining each new vertex to the end vertices of the corresponding edge. These results are then used to construct infinitely many pairs of adjacency (respectively, Laplacian and signless Laplacian) cospectral graphs.

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TL;DR: In this article, the reverse Heinz-type inequalities involving Hadamard product of the formin which is a unitarily invariant norm were established for positive definite matrices.

Abstract: Let be positive definite matrices. We present several reverse Heinz-type inequalities, in particularwhere is an arbitrary matrix, is Hilbert–Schmidt norm and . We also establish a Heinz-type inequality involving Hadamard product of the formin which and is a unitarily invariant norm.

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TL;DR: In this article, a sharp lower bound as well as a sharp upper bound in terms of the clique number were derived for the distance signless Laplacian spectral radius of a connected graph.

Abstract: The distance signless Laplacian spectral radius of a connected graph , denoted by , is the maximal eigenvalue of the distance signless Laplacian matrix of . In this paper, we find a sharp lower bound as well as a sharp upper bound of in terms of the clique number. Furthermore, both extremal graphs are uniquely determined.

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TL;DR: In this paper, the Laplacian spectrum of the power graph of the additive cyclic group and the dihedral group was studied and shown to be the union of that of and.

Abstract: The power graph of a finite group G is the graph whose vertices are the elements of G and two distinct vertices are adjacent if and only if one is an integral power of the other. In this paper, we study Laplacian spectrum of the power graph of additive cyclic group and the dihedral group . We show that the Laplacian spectrum of is the union of that of and . We find algebraic connectivity of and give bounds of the same for .

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TL;DR: In this paper, the star, the left-star and the right-star partial orders of a ring admitting involution are studied and conditions under which these orders are equivalent to the minus partial order are obtained.

Abstract: Let be a unital ring admitting involution. We introduce an order on and show that in the case when is a Rickart -ring, this order is equivalent to the well-known star partial order. The notion of the left-star and the right-star partial orders is extended to Rickart -rings. Properties of the star, the left-star and the right-star partial orders are studied in Rickart -rings and some known results are generalized. We found matrix forms of elements and when , where is one of these orders. Conditions under which these orders are equivalent to the minus partial order are obtained. The diamond partial order is also investigated.

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Hebei University

^{1}TL;DR: In this article, the 3-Lie algebras are constructed by commutative associative algesbras, involutions and derivations, and their structures are studied.

Abstract: 3-Lie algebras are constructed by commutative associative algebras, involutions and derivations. Then the 3-Lie algebras are obtained from Lie algebras and linear functions, and from group algebras F[G] of an abelian group G and homomorphisms . At the end of the paper, the 3-Lie algebras are obtained from Laurent polynomials , and whose structures are studied.

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TL;DR: Using the maximal left ring of quotients, the authors generalized the result on functional identity in triangular rings to the case of functional identity on commuting additive maps and generalized inner biderivations of triangular rings.

Abstract: Using the notion of the maximal left ring of quotients, our recent result on the solutions of functional identity in triangular rings is generalized. Consequently, generalizations of known results on commuting additive maps and generalized inner biderivations of triangular rings are obtained.

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TL;DR: In this paper, the authors established new relations between orthogonal pairs of such matrices lying on the boundary of either cone and established an improvement on the upper bound of the cp-rank of completely positive matrices of general order.

Abstract: Copositive and completely positive matrices play an increasingly important role in Applied Mathematics, namely as a key concept for approximating NP-hard optimization problems. The cone of copositive matrices of a given order and the cone of completely positive matrices of the same order are dual to each other with respect to the standard scalar product on the space of symmetric matrices. This paper establishes some new relations between orthogonal pairs of such matrices lying on the boundary of either cone. As a consequence, we can establish an improvement on the upper bound of the cp-rank of completely positive matrices of general order and a further improvement for such matrices of order six.

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TL;DR: In this article, the -circulant matrix associated with the numbers defined by the recursive relation with initial conditions and, where and, are obtained some formulas for the determinants and inverses of.

Abstract: Let be the -circulant matrix associated with the numbers defined by the recursive relation with initial conditions and , where and We obtain some formulas for the determinants and inverses of . Some bounds for spectral norms of are obtained as applications.

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TL;DR: In this paper, it was shown that a complete graph with a few edges removed is uniquely determined by its permanental spectrum, i.e., if it induces a star or a matching, or a disjoint union of a matching and a path.

Abstract: Let denote the set of all graphs obtained from by removing five or fewer edges. Camara and Haemers proved that graphs in are uniquely determined by their adjacency spectra with the exception for graphs and . In this paper, we show that all graphs in are uniquely determined by their permanental spectra. We further extend our findings by investigating when a complete graph with a few edges removed is uniquely determined by its permanental spectrum. More precisely, we prove that if induces a star, or a matching, or a disjoint union of a matching and a path , then is uniquely determined by its permanental spectrum.

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TL;DR: The algebraic structure of -additive codes which are -submodules where is prime, and and are positive integers is studied, leading to the identification of the type and the cardinalities of these codes.

Abstract: In this paper, we study the algebraic structure of -additive codes which are -submodules where is prime, and and are positive integers. -additive codes naturally generalize and -additive codes which have been introduced recently. The results obtained in this work generalize a great amount of the studies done on additive codes. Especially, we determine the standard forms of generator and parity-check matrices for this family of codes. This leads to the identification of the type and the cardinalities of these codes. Furthermore, we present some bounds on the minimum distance and examples that attain these bounds. Finally, we present some illustrative examples for some special cases.

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TL;DR: In this article, the structure of algebraic morphisms and isometries of the space of all complex positive definite matrices was determined and used to describe all continuous Jordan triple endomorphisms of which are continuous maps satisfying

Abstract: In this paper, we determine the structure of certain algebraic morphisms and isometries of the space of all complex positive definite matrices. In the case , we describe all continuous Jordan triple endomorphisms of which are continuous maps satisfying It has recently been discovered that surjective isometries of certain substructures of groups equipped with metrics which are in a way compatible with the group operations have algebraic properties that relate them rather closely to Jordan triple morphisms. This makes us possible to use our structural results to describe all surjective isometries of that correspond to any member of a large class of metrics generalizing the geodesic distance in the natural Riemannian structure on . Finally, we determine the isometry group of relative to a very recently introduced metric that originates from the divergence called Stein’s loss.

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TL;DR: In this article, the spectrum of the all-one hypermatrix was computed using the Poisson product formula, which includes a complete description of the eigenvalues' multiplicities, a seemingly elusive aspect of the spectral theory of tensors.

Abstract: We compute the spectrum of the ‘all ones’ hypermatrix using the Poisson product formula. This computation includes a complete description of the eigenvalues’ multiplicities, a seemingly elusive aspect of the spectral theory of tensors. We also give a distributional picture of the spectrum as a point-set in the complex plane. Finally, we use the technique to analyse the spectrum of ‘sunflower hypergraphs’, a class that has played a prominent role in extremal hypergraph theory.

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TL;DR: In this paper, lower and upper bounds for the distance Estrada index of a simple graph were established for almost all graphs, and the eigenvalues of its distance matrix were derived.

Abstract: Suppose is a simple graph and are the eigenvalues of its distance matrix . The distance Estrada index of is defined as the sum of , . In this paper, we establish better lower and upper bounds to for almost all graphs .

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TL;DR: In this article, the authors investigated linear split quaternionic equations with the terms of the form axb and gave a new method of solving general linear split quadratic equations with one, two and n unknowns.

Abstract: In this paper, we investigate linear split quaternionic equations with the terms of the form axb. We give a new method of solving general linear split quaternionic equations with one, two and n unknowns. Moreover, we present some examples to show how this procedure works.

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TL;DR: In this paper, it was shown that if the product is generalized Drazin invertible for arbitrary, then the product of the powers of the products of elements and the elements of the elements is invertable.

Abstract: Liao et al. proved that if the product is generalized Drazin invertible, then so is extending the Cline’s formula to the case of the generalized Drazin invertibility. In this paper, we show that if is generalized Drazin invertible for arbitrary , then is generalized Drazin invertible in a Banach algebra. So, we generalize Cline’s formula to the case of the generalized Drazin invertibility of the powers of the products of elements and . Also, we prove that is generalized Drazin invertible if and only if is generalized Drazin invertible; and is generalized Drazin invertible if and only if is generalized Drazin invertible.

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TL;DR: In this article, the authors studied the left and right generalized Drazin inverse of bounded operators in a Banach space and showed that these sets are compact in the complex plane and invariant under additive commuting quasi-nilpotent perturbations.

Abstract: In this paper, we define and study the left and the right generalized Drazin inverse of bounded operators in a Banach space We show that the left (resp the right) generalized Drazin inverse is a sum of a left invertible (resp a right invertible) operator and a quasi-nilpotent one In particular, we define the left and the right generalized Drazin spectra of a bounded operator and also show that these sets are compact in the complex plane and invariant under additive commuting quasi-nilpotent perturbations Furthermore, we prove that a bounded operator is left generalized Drazin invertible if and only if its adjoint is right generalized Drazin invertible An equivalent definition of the pseudo-Fredholm operators in terms of the left generalized Drazin invertible operators is also given Our obtained results are used to investigate some relationships between the left and right generalized Drazin spectra and other spectra founded in Fredholm theory