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Inversion formulas for Toeplitz-plus-Hankel matrices Linear Algebra its Appl. (IF 1.1) Pub Date : 2024-03-08 Torsten Ehrhardt, Karla Rost
The main aim of the present paper is to establish inversion formulas of Gohberg-Semencul type for Toeplitz-plus-Hankel matrices. In particular, it is shown how the inverse of such a structured matrix of order is computed by means of their first two and last two columns or rows under the additional assumption that a certain matrix is nonsingular. Moreover, a formula for the inverse of the Toeplitz-plus-Hankel
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Equivalence for flag codes Linear Algebra its Appl. (IF 1.1) Pub Date : 2024-03-07 Miguel Ángel Navarro-Pérez, Xaro Soler-Escrivà
Given a finite field and a positive integer , a is a sequence of nested -subspaces of a vector space and a is a nonempty collection of flags. The of a flag code are the constant dimension codes containing all the subspaces of prescribed dimensions that form the flags in the flag code.
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On unitary algebras with graded involution of quadratic growth Linear Algebra its Appl. (IF 1.1) Pub Date : 2024-03-05 D.C.L. Bessades, W.D.S. Costa, M.L.O. Santos
Let be a field of characteristic zero. By a ⁎-superalgebra we mean an algebra with graded involution over . Recently, algebras with graded involution have been extensively studied in PI-theory and the sequence of ⁎-graded codimensions has been investigated by several authors. In this paper, we classify varieties generated by unitary ⁎-superalgebras having quadratic growth of ⁎-graded codimensions.
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Distance-regular graphs with exactly one positive q-distance eigenvalue Linear Algebra its Appl. (IF 1.1) Pub Date : 2024-03-05 Jack H. Koolen, Mamoon Abdullah, Brhane Gebremichel, Sakander Hayat
In this paper, we study the -distance matrix for a distance-regular graph and show that the -distance matrix of a distance-regular graph with classical parameters has exactly three distinct eigenvalues, of which one is zero. Moreover, we study distance-regular graphs whose -distance matrix has exactly one positive eigenvalue.
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Bounded rank perturbations of a regular matrix pencil Linear Algebra its Appl. (IF 1.1) Pub Date : 2024-03-02 Marija Dodig, Marko Stošić
In this paper we study the possible Kronecker invariants of an arbitrary matrix pencil obtained by bounded rank perturbation of a regular matrix pencil. We solve this problem in the case of a perturbation of minimal rank.
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The unique spectral extremal graph for intersecting cliques or intersecting odd cycles Linear Algebra its Appl. (IF 1.1) Pub Date : 2024-02-29 Lu Miao, Ruifang Liu, Jingru Zhang
The -fan, denoted by , is the graph consisting of copies of the complete graph which intersect in a single vertex. Desai et al. proved that for sufficiently large , where and are the sets of -vertex -free graphs with maximum spectral radius and maximum size, respectively. In this paper, the set is uniquely determined for large enough.
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Spectral radius of graphs of given size with forbidden subgraphs Linear Algebra its Appl. (IF 1.1) Pub Date : 2024-02-29 Yuxiang Liu, Ligong Wang
Let be the spectral radius of a graph with edges. Let be the graph obtained from by adding disjoint edges within its independent set. Nosal's theorem states that if , then contains a triangle. Zhai and Shu showed that any non-bipartite graph with and contains a quadrilateral unless M.Q. Zhai and J.L. Shu (2022) . Wang proved that if for a graph with size , then contains a quadrilateral unless is one
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Existence of the map [formula omitted] Linear Algebra its Appl. (IF 1.1) Pub Date : 2024-02-29 Steven R. Lippold, Mihai D. Staic
In this paper we show the existence of a nontrivial linear map with the property that if there exists such that . This gives a partial answer to a conjecture from . As an application, we use the map to study those -partitions of the complete hypergraph that have zero Betti numbers. We also discuss algebraic and combinatorial properties of a map which generalizes the determinant map, the map from ,
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Drazin and group invertibility in algebras spanned by two idempotents Linear Algebra its Appl. (IF 1.1) Pub Date : 2024-02-29 Rounak Biswas, Falguni Roy
For two given idempotents from an associative algebra , in this paper, we offer a comprehensive classification of algebras spanned by the idempotents . This classification is based on the condition that are not tightly coupled and satisfy but for some . Subsequently, we categorize all the group invertible elements and establish an upper bound for the Drazin index of any elements in these algebras spanned
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Four-vertex quivers supporting twisted graded Calabi–Yau algebras Linear Algebra its Appl. (IF 1.1) Pub Date : 2024-02-28 Jason Gaddis, Thomas Lamkin, Thy Nguyen, Caleb Wright
We study quivers supporting twisted graded Calabi–Yau algebras. Specifically, we classify quivers on four vertices in which the Nakayama automorphism acts on the degree zero part by either a four-cycle, a three-cycle, or two two-cycles. In order to realize algebras associated to some of these quivers, we show that a graded twist of a twisted graded Calabi–Yau algebra is another algebra of the same
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On connected T-gain graphs with rank equal to girth Linear Algebra its Appl. (IF 1.1) Pub Date : 2024-02-27 Suliman Khan
Let be a -gain graph or a complex unit gain graph and be its adjacency matrix. The rank of is denoted by which is the rank of its adjacency matrix. If the underlying graph Γ of Φ has at least one cycle, then the girth of Φ is denoted by or simply by , which is the length of the shortest cycle in Γ. In this paper, we prove for a -gain graph . Moreover, we characterize -gain graphs satisfy and .
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Partial isospectrality of a matrix pencil and circularity of the c-numerical range Linear Algebra its Appl. (IF 1.1) Pub Date : 2024-02-23 Alma van der Merwe, Madelein van Straaten, Hugo J. Woerdeman
We study when functions of the eigenvalues of the pencil are constant functions of . The results are then applied to questions regarding the numerical range, the higher rank numerical range and the -numerical range, and we derive trace type conditions for when these numerical ranges are disks centered at 0. The theory of symmetric polynomials plays an important part in the proofs.
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The Pascal functional matrices with multidimensional binomial coefficients Linear Algebra its Appl. (IF 1.1) Pub Date : 2024-02-23 Morteza Bayat
In this paper, we generalize the Pascal matrices presented in as a functional for several variables. The entries of these matrices are defined on the set with multidimensional binomial coefficients along with several variables. In the following, we will investigate the algebraic properties of these matrices in terms of products, powers, and inverses of the matrices and also some factorization formulas
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Unique matrix factorizations associated to bilinear forms and Schur multipliers Linear Algebra its Appl. (IF 1.1) Pub Date : 2024-02-23 Erik Christensen
Grothendieck's inequalities for operators and bilinear forms imply some factorization results for complex matrices. The theory of operator spaces provides a set up which describes 4 norm optimal factorizations of Grothendieck's sort. It is shown that 3 of the optimal factorizations are uniquely determined and the remaining one is unique in some cases.
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First order structure-preserving perturbation theory for eigenvalues of symplectic matrices: Part II Linear Algebra its Appl. (IF 1.1) Pub Date : 2024-02-23 Fredy Sosa, Julio Moro, Christian Mehl
The first order eigenvalue perturbation theory for structure-preserving perturbations of real or complex symplectic matrices started in is continued and completed. While in all eigenvalues were considered that qualitatively behave similarly under structure-preserving and structure-ignoring perturbations, the focus of this paper is on one particular case when the behavior under structure-preserving
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Quadrature methods for singular integral equations of Mellin type based on the zeros of classical Jacobi polynomials, II Linear Algebra its Appl. (IF 1.1) Pub Date : 2024-02-21 Peter Junghanns, Robert Kaiser
With this paper we continue the investigations started in and concerned with stability conditions for collocation-quadrature methods based on the zeros of classical Jacobi polynomials, not only Chebyshev polynomials. While in we only proved the necessity of certain conditions, here we will show also their sufficiency in particular cases.
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A spectral dichotomy for commuting m-isometries with negative core operator Linear Algebra its Appl. (IF 1.1) Pub Date : 2024-02-21 Santu Bera, Sameer Chavan, Soumitra Ghara
Let be a pair of commuting operators such that the Taylor spectrum of is contained in the closed unit bidisc and the left spectrum of either of or is contained in the unit circle . If the core operator of is negative, then we show that is either contained in or equal to . This fact applies in particular to a commuting pair of -isometries with negative core operator. Our method of proof relies on a
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Strong star complements in graphs Linear Algebra its Appl. (IF 1.1) Pub Date : 2024-02-21 Milica Anđelić, Peter Rowlinson, Zoran Stanić
Let be a finite simple graph with as an eigenvalue (i.e. an eigenvalue of the adjacency matrix of ), and let be a star complement for in . Motivated by a controllability condition, we say that is a star complement for if and have no eigenvalue in common. We explore this concept in the context of line graphs, exceptional graphs, strongly regular graphs and graphs with a prescribed star complement.
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Cyclic algebras, symbol algebras and gradings on matrices Linear Algebra its Appl. (IF 1.1) Pub Date : 2024-02-21 C. Boboc, S. Dăscălescu, L. van Wyk
We consider cyclic algebras, Milnor's symbol algebras, and certain graded algebra structures on them. We classify these gradings with respect to both isomorphism and equivalence relations. Some of them induce gradings on matrix algebras, which we also classify. As an application, we obtain the classification of all group gradings on the algebra , where is a prime number and is an arbitrary field.
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Co-eigenvector graphs Linear Algebra its Appl. (IF 1.1) Pub Date : 2024-02-20 Piet Van Mieghem, Ivan Jokić
Except for the empty graph, we show that the orthogonal matrix of the adjacency matrix determines that adjacency matrix completely, but not always uniquely. The proof relies on interesting properties of the Hadamard product . As a consequence of the theory, we show that irregular co-eigenvector graphs exist only if the number of nodes . Co-eigenvector graphs possess the same orthogonal eigenvector
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A Riordan group poset Linear Algebra its Appl. (IF 1.1) Pub Date : 2024-02-20 Louis W. Shapiro, Minho Song
In this paper, we look at Riordan arrays with nonnegative integer entries. In the set, if for a nonnegative integer matrix , then we say . Our primary focus centers around six kinds of key arrays: Appell, Bell, two-Bell, Natural, Hankel, and Pseudo-involution. We examine relations among Hankel, Bell and two-Bell in general, while the other entries are treated case by case. We start by looking at individual
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The Frobenius distances from projections to an idempotent matrix Linear Algebra its Appl. (IF 1.1) Pub Date : 2024-02-15 Xiaoyi Tian, Qingxiang Xu, Chunhong Fu
For each pair of matrices and with the same order, let denote their Frobenius distance. This paper deals mainly with the Frobenius distances from projections to an idempotent matrix. For every idempotent , a projection called the matched projection can be induced. It is proved that is the unique projection whose Frobenius distance away from takes the minimum value among all the Frobenius distances
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On linear non-uniform cellular automata: Duality and dynamics Linear Algebra its Appl. (IF 1.1) Pub Date : 2024-02-15 Xuan Kien Phung
For linear non-uniform cellular automata (NUCA) which are global perturbations of CA over an arbitrary universe, we introduce and investigate their dual linear NUCA, which are also endomorphisms over a generally infinite dimensional vector space. Generalizing results for linear CA, we show that dynamical properties namely pre-injectivity, resp. injectivity, resp. stably injectivity, resp. invertibility
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On the distance to low-rank matrices in the maximum norm Linear Algebra its Appl. (IF 1.1) Pub Date : 2024-02-15 Stanislav Budzinskiy
Every sufficiently big matrix with small spectral norm has a nearby low-rank matrix if the distance is measured in the maximum norm (Udell & Townsend, SIAM J Math Data Sci, 2019). We use the Hanson–Wright inequality to improve the estimate of the distance for matrices with incoherent column and row spaces. In numerical experiments with several classes of matrices we study how well the theoretical upper
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Graded identities with involution for the algebra of upper triangular matrices Linear Algebra its Appl. (IF 1.1) Pub Date : 2024-02-15 Diogo Diniz, Alex Ramos, José Lucas Galdino
Let be a field of characteristic zero and let be an integer. In this paper, we prove that if a group grading on admits a graded involution then this grading is a coarsening of a -grading on and the graded involution is equivalent to the reflection or symplectic involution on , this grading is called the finest grading on . Furthermore, if the algebra with the finest grading satisfies no non-trivial
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Functors between representation categories. Universal modules Linear Algebra its Appl. (IF 1.1) Pub Date : 2024-02-15 A.L. Agore
Let and be two Lie algebras with finite dimensional and consider to be the corresponding universal algebra as introduced in . Given an -module and a Lie -module we show that can be naturally endowed with a Lie -module structure. This gives rise to a functor between the category of Lie -modules and the category of Lie -modules and, respectively, to a functor between the category of -modules and the
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Advances on similarity via transversal intersection of manifolds Linear Algebra its Appl. (IF 1.1) Pub Date : 2024-02-12 Marina Arav, Frank J. Hall, Hein van der Holst, Zhongshan Li, Aram Mathivanan, Jiamin Pan, Hanfei Xu, Zheng Yang
Let be an real matrix. As shown in the recent paper S.M. Fallat, H.T. Hall, J.C.-H. Lin, and B.L. Shader (2022) , if the manifolds and (consisting of all real matrices having the same sign pattern as ), both considered as embedded submanifolds of , intersect transversally at , then every superpattern of sgn() also allows a matrix similar to . Those authors introduced a condition on (in terms of certain
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The β maps: Strong clustering and distribution results on the complex unit circle Linear Algebra its Appl. (IF 1.1) Pub Date : 2024-02-12 Alec J.A. Schiavoni-Piazza, David Meadon, Stefano Serra-Capizzano
In the current work, we study the eigenvalue distribution results of a class of non-normal matrix-sequences which may be viewed as a low rank perturbation, depending on a parameter , of the basic Toeplitz matrix-sequence , . The latter of which has obviously all eigenvalues equal to zero for any matrix order , while for the matrix-sequence under consideration we will show a strong clustering on the
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Combinatorial Fiedler theory and graph partition Linear Algebra its Appl. (IF 1.1) Pub Date : 2024-02-09 Enide Andrade, Geir Dahl
Partition problems in graphs are extremely important in applications, as shown in the Data Science and Machine Learning literature. One approach is spectral partitioning based on a Fiedler vector, i.e., an eigenvector corresponding to the second smallest eigenvalue of the Laplacian matrix of the graph . This problem corresponds to the minimization of a quadratic form associated with , under certain
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Preface to the Proceedings of the 24th ILAS Conference, Galway, Ireland, June 2022 Linear Algebra its Appl. (IF 1.1) Pub Date : 2024-02-09 Nicolas Gillis, Rachel Quinlan, Clément de Seguins Pazzis, Helena Šmigoc
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Quantitatively Hyper-Positive Real Rational Functions II Linear Algebra its Appl. (IF 1.1) Pub Date : 2024-02-08 Daniel Alpay, Izchak Lewkowicz
Colloquially, quantitatively Hyper-Positive functions form a sub-family of the “Strictly Positive real” functions, where in addition state-space realization always exists and the limit at infinity is non-singular.
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Spectral properties of token graphs Linear Algebra its Appl. (IF 1.1) Pub Date : 2024-02-08 Sasmita Barik, Piyush Verma
Let be a graph on vertices. For a given integer such that , the -token graph of is defined as the graph whose vertices are the -subsets of the vertex set of , and two of them are adjacent whenever their symmetric difference is a pair of adjacent vertices in . In this article, we study the structural and spectral properties of token graphs. We describe the adjacency matrix and the Laplacian matrix of
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The condition number of singular subspaces, revisited Linear Algebra its Appl. (IF 1.1) Pub Date : 2024-02-08 Nick Vannieuwenhoven
I revisit the condition number of computing left and right singular subspaces from J.-G. Sun (1996) . For real and complex matrices, I present an alternative computation of this condition number in the Euclidean distance on the input space of matrices and the chordal, Grassmann, and Procrustes distances on the output Grassmannian manifold of linear subspaces. Up to a small factor, this condition number
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Geometric and spectral analysis on weighted digraphs Linear Algebra its Appl. (IF 1.1) Pub Date : 2024-02-08 Fernando Lledó, Ignacio Sevillano
In this article we give a geometrical description of the (in general non-selfadjoint) in/out Laplacian and adjacency matrix on digraphs with arbitrary weights, where is the adjoint of the evaluation map on the terminal/initial vertex of each arc and denotes the discrete gradient. We prove that the multiplicity of the zero eigenvalue of coincides with the number of sources/sinks of the digraph. We also
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Some properties concerning Perron vectors of weakly irreducible nonnegative tensors, and their application to rigorous enclosure Linear Algebra its Appl. (IF 1.1) Pub Date : 2024-02-07 Shinya Miyajima
We clarify some properties concerning Perron vectors of weakly irreducible nonnegative (WIN) tensors, and multi-linear systems with non-homogeneous left-hand side whose solutions are sub-vectors of the Perron vectors. Based on the properties, we propose an algorithm for computing interval vectors containing the Perron vectors. This algorithm is applicable to all the WIN tensors, whereas the algorithms
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On polynomial solutions of certain finite order ordinary differential equations Linear Algebra its Appl. (IF 1.1) Pub Date : 2024-02-05 L.M. Anguas, D. Barrios Rolanía
Given a finite order differential operator, some properties and relations satisfied by its polynomial eigenfunctions are studied. Under certain restrictions, such eigenfunctions are explicitly obtained, as well as the corresponding eigenvalues. Also, some linear transformations are applied to sequences of eigenfunctions and a necessary condition for this to be a sequence of eigenfunctions of a new
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The change of vertex energy when joining trees Linear Algebra its Appl. (IF 1.1) Pub Date : 2024-02-05 Octavio Arizmendi, Saylé Sigarreta
In this manuscript we study how the vertex energy of a tree is affected when joined with a bipartite graph. We find an alternating pattern with respect to the coalescence vertex: the energy decreases for vertices located at odd distances and increases for those located at even distances.
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On the ρ-operator radii Linear Algebra its Appl. (IF 1.1) Pub Date : 2024-02-05 Fuad Kittaneh, Ali Zamani
Let and be the operator radius of a bounded linear Hilbert space operator . In this paper we present characterizations of operators satisfying . We also give an expression for the -radii based on the numerical radius of a certain block matrix. This enables us to investigate the properties of the operator radii . In particular, we obtain lower and upper bounds for the operator radii. In addition, we
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An atomic viewpoint of the TP completion problem Linear Algebra its Appl. (IF 1.1) Pub Date : 2024-02-02 Daniel Carter, Charles Johnson
We present two complementary techniques called catalysis and inhibition which allow one to determine if a given pattern is TP completable or TP non-completable, respectively. Empirically, these techniques require considering only one unspecified entry at a time in a vast majority of cases, which makes these techniques ripe for automation and a powerful framework for future work in the TP completion
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On a family of low-rank algorithms for large-scale algebraic Riccati equations Linear Algebra its Appl. (IF 1.1) Pub Date : 2024-02-02 Christian Bertram, Heike Faßbender
In it was shown that four seemingly different algorithms for computing low-rank approximate solutions to the solution of large-scale continuous-time algebraic Riccati equations (CAREs) generate the same sequence when used with the same parameters. The Hermitian low-rank approximations are of the form , where is a matrix with only few columns and is a small square Hermitian matrix. Each generates a
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Linear maps preserving parallel matrix pairs with respect to the Ky-Fan k-norm Linear Algebra its Appl. (IF 1.1) Pub Date : 2024-02-01 Bojan Kuzma, Chi-Kwong Li, Sushil Singla, Edward Poon
Two bounded linear operators and are parallel with respect to a norm if for some scalar with . Characterization is obtained for bijective linear maps sending parallel bounded linear operators to parallel bounded linear operators with respect to the Ky-Fan -norms.
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On generalizing trace minimization principles, II Linear Algebra its Appl. (IF 1.1) Pub Date : 2024-02-01 Xin Liang, Ren-Cang Li
This paper is concerned with establishing a trace minimization principle for two Hermitian matrix pairs. Specifically, we will answer the question: when is subject to (the identity matrix of apt size) finite? Sufficient and necessary conditions are obtained and, when the infimum is finite, an explicit formula for it is established in terms of the finite eigenvalues of the matrix pairs. Our results
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A matrix equation and the Jordan canonical form Linear Algebra its Appl. (IF 1.1) Pub Date : 2024-02-01 Chi-Kwong Li
For a given complex matrix , a necessary and sufficient condition is obtained for the existence of a matrix satisfying , here denotes the Jordan block of 0. An easy construction of the solution is given if it exists. These results lead to a proof of the fact that a nilpotent matrix is similar to a direct sum of Jordan blocks.
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A numerical radius inequality for sector operators Linear Algebra its Appl. (IF 1.1) Pub Date : 2024-02-01 Stephen Drury
We present an optimal upper bound for the operator norm of a sectorial Hilbert space operator in terms of its numerical radius.
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Quantitative version of Gordon's lemma for Hamiltonian with finite range Linear Algebra its Appl. (IF 1.1) Pub Date : 2024-01-22 Licheng Fang, Shuzheng Guo, Yaqun Peng, Fengpeng Wang
In this paper, we investigate a generalization of Schrödinger operator, known as Hamiltonian with finite range. We prove the related Gordon's lemma with quantitative conditions which means the absence of point spectrum.
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Matrix Factorizations for the Generalized Charlier and Meixner Orthogonal Polynomials Linear Algebra its Appl. (IF 1.1) Pub Date : 2024-01-19 Itsaso Fernández-Irisarri, Manuel Mañas
The Cholesky factorization of the moment matrix is considered for the generalized Charlier and generalized Meixner discrete orthogonal polynomials. For the generalized Charlier, we present an alternative derivation of the Laguerre–Freud relations found by Smet and Van Assche. Third-order and second-order nonlinear ordinary differential equations are found for the recursion coefficient γn, that happen
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Power set of some quasinilpotent weighted shifts on ℓp Linear Algebra its Appl. (IF 1.1) Pub Date : 2024-01-19 ChaoLong Hu, YouQing Ji
For a quasinilpotent operator T on a Banach space X, Douglas and Yang defined kx=limsupz→0ln‖(z−T)−1x‖ln‖(z−T)−1‖ for each nonzero vector x∈X, and call Λ(T)={kx:x≠0} the power set of T. They proved that the power set have a close link with T's lattice of hyperinvariant subspaces. In this paper, we compute the power set of some weighted shifts on ℓp for 1≤p<∞. The following results are obtained: (1)
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Adjacency preserving maps between exterior powers of vector spaces Linear Algebra its Appl. (IF 1.1) Pub Date : 2024-01-17 Jinting Lau, Ming Huat Lim
In this paper, a structural characterization of adjacency preserving surjective maps between exterior powers of arbitrary vector spaces is established, which generalizes the fundamental theorem of the geometry of alternate matrices.
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Lipschitz stability of some canonical Jordan bases of real H-selfadjoint matrices under small perturbations Linear Algebra its Appl. (IF 1.1) Pub Date : 2024-01-17 S. Dogruer Akgul, A. Minenkova, V. Olshevsky
In this paper we prove that the new Jordan bases called γ-flipped orthogonal conjugate symmetric (γ-FOCS) are Lipschitz stable under small perturbations as well. We also establish the Lipschitz stability for the classical real canonical Jordan bases.
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Almost Moore and the largest mixed graphs of diameters two and three Linear Algebra its Appl. (IF 1.1) Pub Date : 2024-01-15 C. Dalfó, M.A. Fiol, N. López
Almost Moore mixed graphs appear in the context of the degree/diameter problem as a class of extremal mixed graphs, in the sense that their order is one unit less than the Moore bound for such graphs. The problem of their existence has been considered just for diameter 2. In this paper, we give a complete characterization of these extremal mixed graphs for diameters 2 and 3. We also derive some optimal
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(m,NA)-Isometries and their harmonious applications to Hilbert-Schmidt operators Linear Algebra its Appl. (IF 1.1) Pub Date : 2024-01-15 Mohamed Amine Aouichaoui
Numerous works have been dedicated to the topic of m-isometries, including [3], [4], [5], [6], [7], [15], [19], [20], [21], [28], [49], [50]. In this article, we introduce the concept of (m,NA)-isometry, where A is a non-zero operator and m is a positive integer, as an extension of the m-isometry class created by J. Alger and M. Stankus in the 1980s. We present some algebraic and spectral characteristics
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Moduli of triples of points in quaternionic hyperbolic geometry Linear Algebra its Appl. (IF 1.1) Pub Date : 2024-01-12 Igor Almeida, Nikolay Gusevskii
In this work, we describe the moduli of triples of points in quaternionic projective space which define uniquely the congruence classes of such triples relative to the action of the isometry group of quaternionic hyperbolic space HQn. To solve this problem, we introduce some basic invariants of triples of points in quaternionic hyperbolic geometry. In particular, we define quaternionic analogues of
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