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Dynamical sampling for the recovery of spatially constant source terms in dynamical systems Linear Algebra its Appl. (IF 1.1) Pub Date : 2024-04-19 A. Aldroubi, R. Díaz Martín, I. Medri
In this paper, we investigate the problem of source recovery in a dynamical system utilizing space-time samples. This is a specific issue within the broader field of dynamical sampling, which involves collecting samples from solutions to a differential equation across both space and time with the aim of recovering critical data, such as initial values, the sources, the driving operator, or other relevant
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Signed graphs and inverses of their incidence matrices Linear Algebra its Appl. (IF 1.1) Pub Date : 2024-04-18 Abdullah Alazemi, Milica Anđelić, Sudipta Mallik
The Laplacian matrix of a signed graph may or may not be invertible. We present a combinatorial formula of the Moore-Penrose inverse of . This is achieved by finding a combinatorial formula for the Moore-Penrose inverse of an incidence matrix of . This work generalizes related known results on incidence and Laplacian matrices of an unsigned graph. Several examples are provided to show the usefulness
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Piecewise scalable frames Linear Algebra its Appl. (IF 1.1) Pub Date : 2024-04-16 Peter G. Casazza, Laura De Carli, Tin T. Tran
In this paper we define “piecewise scalable frames”. This new scaling process allows us to alter many frames to Parseval frames which is impossible by the previous standard scaling. We give necessary and sufficient conditions for a frame to be piecewise scalable. We show that piecewise scalability is preserved under unitary transformations. Unlike standard scaling, we show that all frames in and are
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Extensions of Yamamoto-Nayak's theorem Linear Algebra its Appl. (IF 1.1) Pub Date : 2024-04-16 Huajun Huang, Tin-Yau Tam
A result of Nayak asserts that exists for each complex matrix , where , and the limit is given in terms of the spectral decomposition. We extend the result of Nayak, namely, we prove that the limit of exists for any complex matrices , , and , where and are nonsingular; the limit is obtained and is independent of . We then provide generalization in the context of real semisimple Lie groups.
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Eigenvalue multiplicity of a graph in terms of the number of external vertices Linear Algebra its Appl. (IF 1.1) Pub Date : 2024-04-13 Dein Wong, Wenhao Zhen, Songnian Xu
The multiplicity of an eigenvalue of a graph is denoted by . In a connected graph with at least two vertices, a vertex is called external if it is not a cut vertex. In a tree, an external vertex is exactly a pendant vertex. Let be the number of external vertices in . In this paper, we prove that for any and characterize the extremal graphs with , which generalizes the main result of Wang et al. [Linear
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Comparison of K-spectral set bounds on norms of functions of a matrix or operator Linear Algebra its Appl. (IF 1.1) Pub Date : 2024-04-12 Anne Greenbaum, Natalie Wellen
We use results in [M. Crouzeix and A. Greenbaum, , SIAM Jour. Matrix Anal. Appl., 40 (2019), pp. 1087-1101] to derive upper bounds on the norm of a function of a matrix or other bounded linear operator based on the infinity-norm of on various regions in the complex plane. We compare these results to those that can be derived from a straightforward application of the Cauchy integral formula by replacing
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Model order reduction for discrete time-delay systems based on Laguerre function expansion Linear Algebra its Appl. (IF 1.1) Pub Date : 2024-04-10 Xiaolong Wang, Kejia Xu, Li Li
We investigate model order reduction for discrete time-delay systems via orthogonal polynomial expansion in the frequency domain. The transfer function of systems is expanded in the framework of Laguerre function basis. We show that Laguerre coefficients of the states satisfy a linear system and reduced models generated by projection methods can preserve some Laguerre coefficients of the original systems
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Towards understanding CG and GMRES through examples Linear Algebra its Appl. (IF 1.1) Pub Date : 2024-04-08 Erin Carson, Jörg Liesen, Zdeněk Strakoš
When the conjugate gradient (CG) method for solving linear algebraic systems was formulated about 70 years ago by Lanczos, Hestenes, and Stiefel, it was considered an iterative process possessing a mathematical finite termination property. With the deep insight of the original authors, CG was placed into a very rich mathematical context, including links with Gauss quadrature and continued fractions
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Diagonal entries of inverses of diagonally dominant matrices Linear Algebra its Appl. (IF 1.1) Pub Date : 2024-04-06 C.R. Johnson, C. Marijuán, M. Pisonero, I.M. Spitkovsky
We consider invertible, row diagonally dominant real matrices and give inequalities on their minors and diagonal entries of their inverses. A very special case is that all diagonal entries of an inverse, of a row stochastic, row diagonally dominant and invertible matrix, are at least 1, with strict inequality at least when the dominance is strict. This was conjectured in international trade theory
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Algebra environments II. Algebra homomorphisms and derivations Linear Algebra its Appl. (IF 1.1) Pub Date : 2024-04-04 Mircea Martin
Algebra environments provide requisites for studying objects of interest in operator algebra theory, group representation theory, spin geometry, Clifford analysis, and several variable operator theory. The concept is analyzed by developing an algebraic geometry approach. Specific algebraic sets, called structure manifolds of algebra environments, and their Zariski tangent spaces are introduced and
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Optimal implementation of quantum gates with two controls Linear Algebra its Appl. (IF 1.1) Pub Date : 2024-04-04 Jens Palsberg, Nengkun Yu
We give a detailed proof of a well-known theorem in quantum computing. The theorem characterizes the number of two-qubit gates that is necessary for implementing three-qubit quantum gates with two controls. For example, the theorem implies that five 2-qubit gates are necessary for implementing the Toffoli gate. No detailed proof was available earlier.
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Separating invariants for two-dimensional orthogonal groups over finite fields Linear Algebra its Appl. (IF 1.1) Pub Date : 2024-04-03 Artem Lopatin, Pedro Antonio Muniz Martins
We described a minimal separating set for the algebra of -invariant polynomial functions of -tuples of two-dimensional vectors over a finite field .
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Centrosymmetric and reverse matrices in bivariate orthogonal polynomials Linear Algebra its Appl. (IF 1.1) Pub Date : 2024-04-03 Cleonice F. Bracciali, Glalco S. Costa, Teresa E. Pérez
We study bivariate orthogonal polynomials associated with an inner product that satisfies a symmetry property such that it is invariant when both variables are interchanged. Under that hypothesis, the structure of the polynomial vectors of the orthogonal polynomial systems is described by using centrosymmetric matrices. Also, we prove that the coefficient matrices of the three-term relations, one for
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The maximum operator distance from an idempotent to the set of projections Linear Algebra its Appl. (IF 1.1) Pub Date : 2024-04-02 Xiaofeng Zhang, Xiaoyi Tian, Qingxiang Xu
For each pair of bounded linear operators and on a Hilbert space , let be their operator distance. Given an idempotent on , consider the operator distances from to all of the projections on . It is proved that there always exists a projection whose operator distance away from takes the maximum value. An example is constructed to show that such a projection may fail to be unique.
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Complementary vanishing graphs Linear Algebra its Appl. (IF 1.1) Pub Date : 2024-04-02 Craig Erickson, Luyining Gan, Jürgen Kritschgau, Jephian C.-H. Lin, Sam Spiro
Given a graph with vertices , we define to be the set of symmetric matrices such that for we have if and only if . Motivated by the Graph Complement Conjecture, we say that a graph is complementary vanishing if there exist matrices and such that . We provide combinatorial conditions for when a graph is or is not complementary vanishing, and we characterize which graphs are complementary vanishing in
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Eigenvector components of symmetric, graph-related matrices Linear Algebra its Appl. (IF 1.1) Pub Date : 2024-03-29 P. Van Mieghem
Although eigenvectors belong to the core of linear algebra, relatively few closed-form expressions exist, which we bundle and discuss here. A particular goal is their interpretation for graph-related matrices, such as the adjacency matrix of an undirected, possibly weighted graph.
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Bidiagonal factorization of the recurrence matrix for the Hahn multiple orthogonal polynomials Linear Algebra its Appl. (IF 1.1) Pub Date : 2024-03-29 Amílcar Branquinho, Juan E.F. Díaz, Ana Foulquié-Moreno, Manuel Mañas
This paper explores a factorization using bidiagonal matrices of the recurrence matrix of Hahn multiple orthogonal polynomials. The factorization is expressed in terms of ratios involving the generalized hypergeometric function and is proven using recently discovered contiguous relations. Moreover, employing the multiple Askey scheme, a bidiagonal factorization is derived for the Hahn descendants,
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Bounds and extremal graphs for the energy of complex unit gain graphs Linear Algebra its Appl. (IF 1.1) Pub Date : 2024-03-29 Aniruddha Samanta, M. Rajesh Kannan
A complex unit gain graph (-gain graph), is a graph where the gain function assigns a unit complex number to each orientation of an edge of and its inverse is assigned to the opposite orientation. The associated adjacency matrix is defined canonically. The energy of a -gain graph Φ is the sum of the absolute values of all eigenvalues of . For any connected triangle-free -gain graph Φ with the minimum
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Polynomial reconstruction problem for hypergraphs Linear Algebra its Appl. (IF 1.1) Pub Date : 2024-03-28 Joshua Cooper, Utku Okur
We show that, in general, the characteristic polynomial of a hypergraph is not determined by its “polynomial deck”, the multiset of characteristic polynomials of its vertex-deleted subgraphs, thus settling the “polynomial reconstruction problem” for hypergraphs in the negative. The proof proceeds by showing that a construction due to Kocay of an infinite family of pairs of 3-uniform hypergraphs which
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Quasi-hereditary rings are closed under taking block extensions Linear Algebra its Appl. (IF 1.1) Pub Date : 2024-03-28 Takahide Adachi, Mayu Tsukamoto
In this paper, we give a sufficient condition for Morita context rings to be quasi-hereditary. As an application, we show that each block extension of a quasi-hereditary ring is also quasi-hereditary.
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Some points of view on Grothendieck's inequalities Linear Algebra its Appl. (IF 1.1) Pub Date : 2024-03-28 Erik Christensen
Haagerup's proof of the raises some questions on the and it offers a new result on scalar matrices with non negative entries. The theory of completely bounded maps may be used to show that the follows from the and that this passage may be given a geometric form as a relation between a pair of compact convex sets of positive matrices, which, in turn, characterizes the
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Waring problem for triangular matrix algebra Linear Algebra its Appl. (IF 1.1) Pub Date : 2024-03-28 Rahul Kaushik, Anupam Singh
The Matrix Waring problem is if we can write every matrix as a sum of -th powers. Here, we look at the same problem for triangular matrix algebra consisting of upper triangular matrices over a finite field. We prove that for all integers , there exists a constant , such that for all , every matrix in is a sum of three -th powers. Moreover, if −1 is -th power in , then for all , every matrix in is a
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On the periodicity of singular vectors and the holomorphic block-circulant SVD on the unit circumference Linear Algebra its Appl. (IF 1.1) Pub Date : 2024-03-28 Giovanni Barbarino
We investigate the singular value decomposition of a rectangular matrix that is analytic on the complex unit circumference, which occurs, e.g., with the matrix of transfer functions representing a broadband multiple-input multiple-output channel. Our analysis is based on the Puiseux series expansion of the eigenvalue decomposition of analytic para-Hermitian matrices on the complex unit circumference
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The maximum four point condition matrix of a tree Linear Algebra its Appl. (IF 1.1) Pub Date : 2024-03-27 Ali Azimi, Rakesh Jana, Mukesh Kumar Nagar, Sivaramakrishnan Sivasubramanian
The Four point condition (4PC henceforth) is a well known condition characterising distances in trees . Let be four vertices in and let denote the distance between vertices in . The 4PC condition says that among the three terms , and the maximum value equals the second maximum value.
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Covariance matrices of length power functionals of random geometric graphs – an asymptotic analysis Linear Algebra its Appl. (IF 1.1) Pub Date : 2024-03-27 Matthias Reitzner, Tim Römer, Mandala von Westenholz
Asymptotic properties of a vector of length power functionals of random geometric graphs are investigated. Algebraic properties of the asymptotic covariance matrix are studied as the intensity of the underlying homogeneous Poisson point process increases. This includes a systematic discussion of matrix properties like rank, definiteness, determinant, eigenspaces or decompositions of interest. For the
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On the spectral radius of graphs with given maximum degree and girth Linear Algebra its Appl. (IF 1.1) Pub Date : 2024-03-27 Jiangdong Ai, Seonghyuk Im, Jaehoon Kim, Hyunwoo Lee, Suil O, Liwen Zhang
We prove upper bounds for the spectral radius of an -vertex graph with given maximum degree and girth at least . This extends the previous result of regarding graphs with girth at least five. When or is relatively small compared with the maximum degree, our upper bounds are sharp. In addition, for a tree , we provide an upper bound for the spectral radius of an -vertex -free graph with given maximum
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On the diagonal product of special unitary matrices Linear Algebra its Appl. (IF 1.1) Pub Date : 2024-03-27 Tomasz Miller
We study the image of under the diagonal product map. Using only elementary tools of linear algebra, analysis and general topology, we find the analytical formula for the boundary of this image and find all special unitary matrices whose diagonal product lies on that boundary.
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GLT sequences and automatic computation of the symbol Linear Algebra its Appl. (IF 1.1) Pub Date : 2024-03-27 N.S. Sarathkumar, S. Serra-Capizzano
Spectral and singular value symbols are valuable tools to analyse the eigenvalue or singular value distributions of matrix-sequences in the Weyl sense. More recently, Generalized Locally Toeplitz (GLT) sequences of matrices have been introduced for the spectral/singular value study of the numerical approximations of differential operators in several contexts. As an example, such matrix-sequences stem
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A characterization of sets realizable by compensation in the SNIEP Linear Algebra its Appl. (IF 1.1) Pub Date : 2024-03-27 Carlos Marijuán, Julio Moro
The symmetric nonnegative inverse eigenvalue problem (SNIEP) is the problem of characterizing all possible spectra of entry-wise nonnegative symmetric matrices of given dimension. A list of real numbers is said to be symmetrically realizable if it is the spectrum of some nonnegative symmetric matrix. One of the most general sufficient conditions for realizability is the so-called C-realizability, which
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Optimal properties related to Total Positivity and Wronskian matrices Linear Algebra its Appl. (IF 1.1) Pub Date : 2024-03-26 J.M. Peña
B-bases are totally positive bases satisfying several optimal properties. In this paper we prove a new optimal property of B-bases related to the conditioning of their Wronskian matrices.
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Lax-type pairs in the theory of bivariate orthogonal polynomials Linear Algebra its Appl. (IF 1.1) Pub Date : 2024-03-25 Amílcar Branquinho, Ana Foulquié-Moreno, Teresa E. Pérez, Miguel A. Piñar
Sequences of bivariate orthogonal polynomials written as vector polynomials of increasing size satisfy a couple of three term relations with matrix coefficients. In this work, introducing a time-dependent parameter, we analyze a Lax-type pair system for the coefficients of the three term relations. We also deduce several characterizations relating the Lax-type pair, the shape of the weight, Stieltjes
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Extremal values for the spectral radius of the normalized distance Laplacian Linear Algebra its Appl. (IF 1.1) Pub Date : 2024-03-25 Jacob Johnston, Michael Tait
The normalized distance Laplacian of a graph is defined as where is the matrix with pairwise distances between vertices and is the diagonal transmission matrix. In this project, we study the minimum and maximum spectral radii associated with this matrix, and the structures of the graphs that achieve these values. In particular, we prove a conjecture of Reinhart that the complete graph is the unique
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Spectral Turán problem for [formula omitted]-free signed graphs Linear Algebra its Appl. (IF 1.1) Pub Date : 2024-03-25 Yongang Wang
The classical spectral Turán problem is to determine the maximum spectral radius of an -free graph of order . Let be the set of all unbalanced . Wang, Hou and Li and Chen and Yuan studied the existence of unbalanced and , respectively. In this paper, we focus on the spectral Turán problem of -free signed graph for . By using a different method, we give an answer for and completely characterize the
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On Rotfel'd type inequalities for sectorial matrices Linear Algebra its Appl. (IF 1.1) Pub Date : 2024-03-24 Yanling Mao, Guoxing Ji
Using a matrix decomposition for sectorial matrices and an AM-GM inequality for singular values, we present a new Rotfel'd type inequality for sectorial matrices, which generalizes and partially improves some results by Zhang. We also give alternative simpler proofs of a main result of a recent paper by Yang, Lu and Chen as well as that by Fu and Liu.
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Banach algebra mappings preserving the invertibility of linear pencils Linear Algebra its Appl. (IF 1.1) Pub Date : 2024-03-24 Francois Schulz
Let and be complex unital Banach algebras, and let be surjective mappings. If is semisimple with an essential socle and and together preserve the invertibility of linear pencils in both directions, that is, for any and , is invertible in if and only if is invertible in , then we show that there exists an invertible element in and a Jordan isomorphism such that for all .
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About the persymmetric nonnegative inverse eigenvalue problem in low dimension Linear Algebra its Appl. (IF 1.1) Pub Date : 2024-03-21 Ana I. Julio, Valeria Collants, Roberto C. Díaz
A list of complex numbers is said to be if there exists an nonnegative matrix whose spectrum is Λ. In this case is called a realizing matrix for Λ. The problem of characterizing all realizable lists Λ is known as the (NIEP). If is required to be , the (PNIEP) arises. In this paper, we show that the NIEP and the PNIEP are equivalent for any realizable list of three complex numbers. Furthermore, we give
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Further results on the concavity of operator means Linear Algebra its Appl. (IF 1.1) Pub Date : 2024-03-20 Marek Niezgoda
In this work, we provide some further refinements for the following concavity inequality for positive invertible bounded linear operators on a complex Hilbert space and for an operator mean . To this end, we use the operator majorization intended for comparing two -tuples of pairs of operators, and prove a Hardy-Littlewood-Pólya-Karamata (HLPK) type theorem for the triangle map induced by . Next, we
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On the uniqueness of correspondence analysis solutions Linear Algebra its Appl. (IF 1.1) Pub Date : 2024-03-19 Rick S.H. Willemsen, Michel van de Velden, Wilco van den Heuvel
In correspondence analysis (CA), the rows and columns of a contingency table are optimally represented in a -dimensional approximation, where it is common to set (which includes a so-called trivial dimension). Since CA is a dimension reduction technique, we might expect that the -dimensional approximation is not unique, i.e. there exist several contingency tables with the same -dimensional approximation
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Gain total graphs and their spectra via G-phases and group representations Linear Algebra its Appl. (IF 1.1) Pub Date : 2024-03-18 Matteo Cavaleri, Alfredo Donno, Stefano Spessato
We introduce a definition for the total graph of a gain graph on a group by using -phases, which are a generalization of the notion of orientation to gain graphs. Our construction is well-defined in the sense that gain graphs that are switching isomorphic have switching isomorphic total graphs. In particular, the switching equivalence class of the total graph does not depend on the particular choice
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Distinct eigenvalues of the Transposition graph Linear Algebra its Appl. (IF 1.1) Pub Date : 2024-03-16 Elena V. Konstantinova, Artem Kravchuk
Transposition graph is defined as a Cayley graph over the symmetric group generated by all transpositions. It is known that all eigenvalues of are integers. Moreover, zero is its eigenvalue for any . But the exact distribution of the spectrum of the graph is unknown. In this paper we prove that integers from the interval lie in the spectrum of for any .
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Deletion-contraction for a unified Laplacian and applications Linear Algebra its Appl. (IF 1.1) Pub Date : 2024-03-16 Farid Aliniaeifard, Victor Wang, Stephanie van Willigenburg
We define a graph Laplacian with vertex weights in addition to the more classical edge weights, which unifies the combinatorial Laplacian and the normalised Laplacian. Moreover, we give a combinatorial interpretation for the coefficients of the weighted Laplacian characteristic polynomial in terms of weighted spanning forests and use this to prove a deletion-contraction relation. We prove various interlacing
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Vanishing immanants Linear Algebra its Appl. (IF 1.1) Pub Date : 2024-03-16 Hassan Cheraghpour, Bojan Kuzma
We classify all the irreducible characters of a symmetric group such that the induced immanant function vanishes identically on alternate matrices with the entries in the complex field.
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The hyperbolic cosine transform and its applications to composition operators Linear Algebra its Appl. (IF 1.1) Pub Date : 2024-03-16 Jan Stochel, Jerzy Bartłomiej Stochel
In this paper we characterize hyperbolic cosine transforms of (positive) Borel measures in terms of exponential convexity (Bernstein's terminology). The case of compactly supported measures is also considered. All of this is then applied to (bounded) composition operators on with affine symbols , where , , is a continuous positive real valued function and is the Euclidean norm on . The main result
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Adjacency preserving maps on symmetric tensors Linear Algebra its Appl. (IF 1.1) Pub Date : 2024-03-16 Wai Leong Chooi, Jinting Lau
Let and be positive integers such that . Let and be vector spaces over fields and , respectively, such that and has at least elements. In this paper, we characterize surjective maps preserving adjacency in both directions on symmetric tensors of finite order, which generalizes Hua's fundamental theorem of geometry of symmetric matrices. We give examples showing the indispensability of the assumptions
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3-Uniform hypergraphs from vector spaces Linear Algebra its Appl. (IF 1.1) Pub Date : 2024-03-15 Jeroen Meulewaeter, Hendrik Van Maldeghem
The Fundamental Theorem of Projective Geometry states that, in a vector space, a permutation of vector lines preserving triples that span a vector plane is induced by a semi-linear automorphism. We consider a generalisation to triples of subspaces, not necessarily of the same dimension, spanning, or being contained in a subspace of fixed dimension. We determine all cases in which the permutation is
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Jordan structures of nilpotent matrices in the centralizer of a nilpotent matrix with two Jordan blocks of the same size Linear Algebra its Appl. (IF 1.1) Pub Date : 2024-03-15 Duško Bogdanić, Alen Đurić, Sara Koljančić, Polona Oblak, Klemen Šivic
In this paper we characterize all nilpotent orbits under the action by conjugation that intersect the nilpotent centralizer of a nilpotent matrix consisting of two Jordan blocks of the same size. We list all the possible Jordan canonical forms of the nilpotent matrices that commute with by characterizing the corresponding partitions.
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Mixed de Branges–Rovnyak and sub-Bergman spaces Linear Algebra its Appl. (IF 1.1) Pub Date : 2024-03-15 Caixing Gu, In Sung Hwang, Woo Young Lee, Jaehui Park
In this paper we obtain some general results about mixed second order de Branges–Rovnyak spaces defined by an operator such that both and are 2-hypercontractions. As applications of these results, we study mixed second order sub-Bergman spaces since the analytic Toeplitz operator on weighted Bergman space for on the unit disk is such that both and are 2-hypercontractions.
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On local preservation of orthogonality and its application to isometries Linear Algebra its Appl. (IF 1.1) Pub Date : 2024-03-15 Debmalya Sain, Jayanta Manna, Kallol Paul
We investigate the local preservation of Birkhoff-James orthogonality at a point by a linear operator on a finite-dimensional Banach space and illustrate its importance in understanding the action of the operator in terms of the geometry of the concerned spaces. In particular, it is shown that such a study is related to the preservation of k-smoothness and the extremal properties of the unit ball of
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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 [formula omitted]-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.