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Linear Algebra and its Applications
Elsevier
Linear Algebra and its Applications

Elsevier

0024-3795

Linear Algebra and its Applications/Journal Linear Algebra and its ApplicationsSCIISTPEIAHCI
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    Involutive random walks on total orders and the anti-diagonal eigenvalue property

    Britnell, John R.Wildon, Mark
    47页
    查看更多>>摘要：This paper studies a family of random walks defined on the finite ordinals using their order reversing involutions. Starting at x is an element of {0, 1, ..., n - 1}, an element y x is chosen according to a prescribed probability distribution, and the walk then steps to n - 1 - y. We show that under very mild assumptions these walks are irreducible, recurrent and ergodic. We then find the invariant distributions, eigenvalues and eigenvectors of a distinguished subfamily of walks whose transition matrices have the global anti-diagonal eigenvalue property studied in earlier work by Ochiai, Sasada, Shirai and Tsuboi. We prove that this subfamily of walks is characterised by their reversibility. As a corollary, we obtain the invariant distributions and rate of convergence of the random walk on the set of subsets of {1, ..., m} in which steps are taken alternately to subsets and supersets, each chosen equiprobably. We then consider analogously defined random walks on the real interval [0, 1] and use techniques from the theory of self-adjoint compact operators on Hilbert spaces to prove analogues of the main results in the discrete case. (C) 2022 Elsevier Inc. All rights reserved.
      原文链接:
    • NSTL
    • Elsevier

    Some families of integral mixed graphs

    Tapia, KatherineAndrade, EnideBonifacio, Andrea SoaresRobbiano, Maria...
    19页
    查看更多>>摘要：A mixed graph (G) over cap is a graph where two vertices can be connected by an edge or by an arc (directed edge). The adjacency matrix, (A) over cap((G) over cap), of a mixed graph has rows and columns indexed by the set of vertices of G, being its {u, v}-entry equal to 1 (respectively, -1) if the vertex u is connected by an edge (respectively, an arc) to the vertex v, and 0 otherwise. These graphs are called integral mixed graphs if the eigenvalues of its adjacency matrix are integers. In this paper, symmetric block circulant matrices are characterized, and as a consequence, the definition of a mixed graph to be a block circulant graph is presented. Moreover, using this concept and the concept of a g-circulant matrix, the construction of a family of undirected graphs that are integral block circulant graphs is shown. These results are extended using the notion of H-join operation to characterize the spectrum of a family of integral mixed graphs. Furthermore, a new binary operation called mixed asymmetric product of mixed graphs is introduced, and the notions of joining by arcs and joining by edges are used, allowing us to obtain a new integral mixed graph from two original integral mixed graphs. (C) 2022 Elsevier Inc. All rights reserved.
      原文链接:
    • NSTL
    • Elsevier

    Exact solutions in low-rank approximation with zeros

    Kubjas, KaieSodomaco, LucaTsigaridas, Elias
    31页
    查看更多>>摘要：Low-rank approximation with zeros aims to find a matrix of fixed rank and with a fixed zero pattern that minimizes the Euclidean distance to a given data matrix. We study the critical points of this optimization problem using algebraic tools. In particular, we describe special linear, affine, and determinantal relations satisfied by the critical points. We also investigate the number of critical points and how this number is related to the complexity of nonnegative matrix factorization problem. (C) 2022 Elsevier Inc. All rights reserved.
      原文链接:
    • NSTL
    • Elsevier

    Symmetric and skew-symmetric polynomial identities with involution for the upper triangular matrix algebras of even order

    Quispe Urure, Ronald Ismael
    17页
    查看更多>>摘要：Let UT2n be the algebra of all 2n x 2n upper triangular matrices over a field F whose characteristic is different from 2. In the present work, we will consider UT2n equipped with an involution of the first kind. In this setting, for certain subalgebras of UT2n (including UT2n itself), we will describe all the *-polynomial identities which are of the form f +/- f* where f is a product of commutators. In addition, we will exhibit sets of linear generators for the relatively free algebras induced from these types of identities. (C) 2022 Elsevier Inc. All rights reserved.
      原文链接:
    • NSTL
    • Elsevier

    Perron-Frobenius theory for some classes of nonnegative tensors in the max algebra

    Khaleghzade, SedigheZangiabadi, MostafaPeperko, AljosaHajarian, Masoud...
    28页
    查看更多>>摘要：An analog of Perron-Frobenius theory is proposed for some classes of nonnegative tensors in the max algebra. In the first part some important characterizations of nonnegative matrices can be extended to nonnegative tensors over max algebra, especially the Perron-Frobenius theorem for weakly irreducible nonnegative tensors and the Collatz-Wielandt minimax theorem for nonnegative tensors. Then, in the second part, an iterative method is proposed for finding the largest max eigenvalue of a nonnegative tensor based on diagonal similar tensors. The iterative method is convergent for weakly irreducible nonnegative tensors. Some numerical results are provided to illustrate the efficiency of the iterative method. (C) 2022 Elsevier Inc. All rights reserved.
      原文链接:
    • NSTL
    • Elsevier

    On automorphism groups of idempotent evolution algebras

    Sriwongsa, SongponZou, Yi Ming
    13页
    查看更多>>摘要：We study the automorphism group of an idempotent evolution algebra, show that any finite group can be the automorphism group of an evolution algebra, and describe certain evolution algebras with given automorphism groups. In particular, we classify n-dimensional idempotent evolution algebras whose automorphism group is isomorphic to the symmetric group S-n, and classify idempotent evolution algebras with maximal diagonal automorphism subgroups. (C) 2022 Elsevier Inc. All rights reserved.
      原文链接:
    • NSTL
    • Elsevier

    On the rank of Hankel matrices over finite fields

    Dwivedi, Omesh DharGrinberg, Darij
    26页
    查看更多>>摘要：Given three nonnegative integers p, q, r and a finite field F, how many Hankel matrices (x(i+j))(0 <= i <= p), (0 <= j <= q) over F have rank <= r? This question is classical, and the answer (q(2r) when r <= min {p, q}) has been obtained independently by various authors using different tools ([3, Theorem 1 for m = n], [4, (26)], [5, Theorem 5.1]). In this note, we will study a refinement of this result: We will show that if we fix the first k of the entries x(0), x(1), ..., x(k-1) for some k <= r <= min {p, q}, then the number of ways to choose the remaining p + q - k + 1 entries x(k), x(k+1), ..., x(p+q) such that the resulting Hankel matrix (x(i+j))(0 <= i <= p), (0 <= j <= q) has rank <= r is q(2r-k). This is exactly the answer thiat one would expect if the first k entries had no effect on the rank, but of course the situation is not this simple (and we had to combine some ideas from [4, (26)] and from [5, Theorem 5.1 for r = n] to obtain our proof). The refined result generalizes (and provides an alternative proof of) [1, Corollary 6.4]. (C) 2022 Elsevier Inc. All rights reserved.
      原文链接:
    • NSTL
    • Elsevier

    Spectral classes of strongly-regular and distance-regular graphs

    Ghorbani, EbrahimKoohestani, Masoumeh
    18页
    查看更多>>摘要：Gu, Jost, Liu, and Stadler (2016) introduced a notion of spectral limit for sequences of graphs. More precisely, this is the limit of the Radon probability measures associated to graphs based on their spectrum of normalized Laplacian matrix. We determine the spectral limit for sequences of strongly-regular as well as distance-regular graphs with classical parameters. (C) 2022 Elsevier Inc. All rights reserved.
      原文链接:
    • NSTL
    • Elsevier

    Counterexamples of the Bhattacharya-Friedland-Peled conjecture

    Cheng, Yen-JenLiu, Chia-AnWeng, Chih-wen
    8页
    查看更多>>摘要：The Brualdi-Hoffman conjecture, proved by Rowlinson in 1988, characterized the graph with maximal spectral radius among all simple graphs with prescribed number of edges. In 2008, Bhattacharya, Friedland, and Peled proposed an analog, which will be called the BFP conjecture in the following, of the Brualdi-Hoffman conjecture for the bipartite graphs with fixed numbers of edges in the graph and vertices in the bipartition. The BFP conjecture was proved to be correct if the number of edges is large enough by several authors. However, in this paper we provide some counterexamples of the BFP conjecture. (C) 2022 Elsevier Inc. All rights reserved.
      原文链接:
    • NSTL
    • Elsevier