查看更多>>摘要：The Hilbert L-matrix A(s)=[a(ij)(s)], where a(ij)(s) = 1/(max{i, j} + s) with i, j >= 0, was introduced in [3]. As a surprising property, we showed that its 2-norm is constant for s >= s(0), where the critical point s(0) is unknown but relies in the interval (1/4, 1/2). In this note, using some delicate calculations we sharpen this result by improving the upper and lower bounds of the interval surrounding s0. Moreover, we establish that the same property persists for the p-norm of A(s) matrices. (C) 2021 Elsevier Inc. All rights reserved.
查看更多>>摘要：This paper concerns matrix decompositions in which the factors are restricted to lie in a closed subvariety of a matrix group. Such decompositions are of relevance in control theory: given a target matrix in the group, can it be decomposed as a product of elements in the subvarieties, in a given order? And if so, what can be said about the solution set to this problem? Can an irreducible curve of target matrices be lifted to an irreducible curve of factorisations? We show that under certain conditions, for a sufficiently long and complicated such sequence, the solution set is always irreducible, and we show that every connected matrix group has a sequence of one-parameter subgroups that satisfies these conditions, where the sequence has length less than 1.5 times the dimension of the group. (C) 2021 The Author(s). Published by Elsevier Inc.
查看更多>>摘要：In 1972, Cvetkovic proved that if G is an n-vertex simple graph with the chromatic number k, then its spectral radius is at most the spectral radius of the n-vertex balanced complete k-partite graph. In this paper, we analyze the characteristic polynomial of a digraph D to prove a tight upper bound for the spectral radius of D in terms of the number of vertices and the chromatic number of D; we also characterize when equality holds. This provides a simple proof of a result by Lin and Shu [6]. (C) 2021 The Authors. Published by Elsevier Inc.
查看更多>>摘要：For an eigenvalue lambda(0) of a Hermitian matrix A, the formula of Thompson and McEnteggert gives an explicit expression of the adjugate of lambda I-0 - A, Adj(lambda I-0 - A), in terms of eigenvectors of Afor lambda(0) and all its eigenvalues. In this paper Thompson-McEnteggert's formula is generalized to include any matrix with entries in an arbitrary field. In addition, for any nonsingular matrix A, a formula for the elementary divisors of Adj(A) is provided in terms of those of A. Finally, a generalization of the eigenvalue-eigenvector identity and three applications of the Thompson-McEnteggert's formula are presented. (C) 2021 The Author(s). Published by Elsevier Inc.
Farenick, DouglasOjo, Oluwatobi RuthPlosker, Sarah
20页
查看更多>>摘要：Weyl's unitary matrices, which were introduced in Weyl's 1927 paper [12] on group theory and quantum mechanics, are p xp unitary matrices given by the diagonal matrix whose entries are the p-th roots of unity and the cyclic shift matrix. Weyl's unitaries, which we denote by uand v, satisfy u(p)= v(p)= 1p( the p xpidentity matrix) and the commutation relation u(v) =zeta vu, where.is a primitive p-th root of unity. We prove that Weyl's unitary matrices are universal in the following sense: if uand vare any d xdunitary matrices such that u(p) = v(p)= 1(d) and uv= sigma vu, then there exists a unital completely positive linear map phi : Mp(C) -> M-d(C) such that f(u) = uand f(v) = v. We also show, moreover, that any two pairs of p-th order unitary matrices that satisfy the Weyl commutation relation are completely order equivalent, but that the assertion for three such unitaries fails. There is a standard tensor-product construction involving the Pauli matrices that produces irreducible sequences of anticommuting selfadjoint unitary matrices of arbitrary length. The matrices in this sequence are called Weyl-Brauer unitary matrices [11, Definition 6.63]. This standard construction is generalised herein to the case p >= 3, producing a sequence of matrices that we also call Weyl-Brauer unitary matrices. We show that the Weyl-Brauer unitary matrices, a g-tuple, are extremal in their matrix range, using recent ideas from noncommutative convexity theory. (C) 2021 Elsevier Inc. All rights reserved.
查看更多>>摘要：An oriented hypergraph is an oriented incidence structure that allows for the generalization of graph theoretic concepts to integer matrices through its locally signed graphic substructure. The locally graphic behaviors are formalized in the subobject classifier of incidence hypergraphs. Moreover, the injective envelope is calculated and shown to contain the class of uniform hypergraphs - providing a combinatorial framework for the entries of incidence matrices. A multivariable all-minors characteristic polynomial is obtained for both the determinant and permanent of the oriented hypergraphic Laplacian and adjacency matrices arising from any integer incidence matrix. The coefficients of each polynomial are shown to be submonic maps from the same family into the injective envelope limited by the subobject classifier. These results provide a unifying theorem for oriented hypergraphic matrix-tree-type and Sachs-coefficient-type theorems. Finally, by specializing to bidirected graphs, the trivial subclasses for the degree-kmonomials of the Laplacian are shown to be in one-to-one correspondence with k-arborescences. (C) 2021 Elsevier Inc. All rights reserved.
查看更多>>摘要：Let R= R(d(t), h(t)) be a Riordan array, where d(t) = Sigma(dn)(n >= 0)t(n) and h(t) = Sigma(n >= 0)h(n)t(n). We show that if the matrix [d(0) h(0) 0 0 ... d(1) h(1) h(0) 0 d(2) h(2) h(1) h(0) . . .. is totally positive, then so is the Riordan array R. (C) 2021 Elsevier Inc. All rights reserved.
查看更多>>摘要：We prove new explicit asymptotic formulae between (geometric) eigenvalues in max-algebra and classical distinguished eigenvalues of nonnegative matrices, which are useful tools for transferring results between both settings. We establish new inequalities for both types of eigenvalues of Hadamard products and Hadamard weighted geometric means of non-negative matrices. Moreover, a version of the spectral mapping theorem for the distinguished spectrum is pointed out. (C) 2021 Elsevier Inc. All rights reserved.
Conde, Cristian M.Dratman, EzequielGrippo, Luciano N.
12页
查看更多>>摘要：Let B(n, q) be the class of block graphs on n vertices having all their blocks of the same size. We prove that if G is an element of B(n, q) has at most three pairwise adjacent cut vertices then the minimum spectral radius rho(G) is attained at a unique graph. In addition, we present a lower bound for rho(G) when G is an element of B(n, q). (C) 2021 Elsevier Inc. All rights reserved.
查看更多>>摘要：We study the closure of the locus of radical ideals in the multigraded Hilbert scheme associated with a standard graded polynomial ring and the Hilbert function of a homogeneous coordinate ring of points in general position in projective space. In the case of projective plane, we give a sufficient condition for an ideal to be in the closure of the locus of radical ideals. For projective space of arbitrary dimension we present a necessary condition. The paper is motivated by the border apolarity lemma which connects such multigraded Hilbert schemes with the theory of ranks of polynomials. (C) 2021 The Author(s). Published by Elsevier Inc.
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