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1.
Let A and B be n?×?n matrices over an algebraically closed field F. The pair ( A,?B ) is said to be spectrally complete if, for every sequence c1,…,cn ∈F such that det (AB)=c1 ,…,cn , there exist matrices A′,B,′∈F,n×n similar to A,?B, respectively, such that A′B′ has eigenvalues c1,…,cn . In this article, we describe the spectrally complete pairs. Assuming that A and B are nonsingular, the possible eigenvalues of A′B′ when A′ and B′ run over the sets of the matrices similar to A and B, respectively, were described in a previous article. 相似文献
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Let c(x 1,?…?,?x d ) be a multihomogeneous central polynomial for the n?×?n matrix algebra M n (K) over an infinite field K of positive characteristic p. We show that there exists a multihomogeneous polynomial c 0(x 1,?…?,?x d ) of the same degree and with coefficients in the prime field 𝔽 p which is central for the algebra M n (F) for any (possibly finite) field F of characteristic p. The proof is elementary and uses standard combinatorial techniques only. 相似文献
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Let X ? denotes the Moore--Penrose pseudoinverse of a matrix X. We study a number of situations when (aA?+?bB)??=?aA?+?bB provided a,?b?∈?????{0} and A, B are n?×?n complex matrices such that A ??=?A and B ??=?B. 相似文献
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Let 𝒯(n,?r;?W n?1) be the set of all n-vertex weighted trees with r vertices of degree 2 and fixed positive weight set W n?1, 𝒫(n,?γ;?W n?1) the set of all n-vertex weighted trees with q pendants and fixed positive weight set W n?1, where W n?1?=?{w 1,?w 2,?…?,?w n?1} with w 1???w 2???···???w n?1?>?0. In this article, we first identify the unique weighted tree in 𝒯(n,?r;?W n?1) with the largest adjacency spectral radius. Then we characterize the unique weighted trees with the largest adjacency spectral radius in 𝒫(n,?γ;?W n?1). 相似文献
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Yiu Tung Poon 《Linear and Multilinear Algebra》2013,61(1-3):221-223
Let A 1,…,Am be nxn hermitian matrices. Definine W(A 1,…,Am )={(xA1x ?,…xAmx ?):x?C n ,xx ?=1}. We will show that every point in the convex hull of W(A 1,…,Am ) can be represented as a convex combination of not more than k(m,n) points in W(A 1,…,Am ) where k(m,n)=min{n,[√m]+δ n 2 m+1}. 相似文献
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Let F be a field and let {d 1,…,dk } be a set of independent indeterminates over F. Let A(d 1,…,dk ) be an n × n matrix each of whose entries is an element of F or a sum of an element of F and one of the indeterminates in {d 1,…,dk }. We assume that no d 1 appears twice in A(d 1,…,dk ). We show that if det A(d 1,…,dk ) = 0 then A(d 1,…,dk ) must contain an r × s submatrix B, with entries in F, so that r + s = n + p and rank B ? p ? 1: for some positive integer p. 相似文献
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Let A be doubly stochastic, and let τ1,…,τm be m mutually disjoint zero diagonals in A, 1?m?n-1. E. T. H. Wang conjectured that if every diagonal in A disjoint from each τk (k=1,…,m) has a constant sum, then all entries in A off the m zero diagonals τk are equal to (n?m)-1. Sinkhorn showed the conjecture to be correct. In this paper we generalize this result for arbitrary doubly stochastic zero patterns. 相似文献
10.
Yongge Tian 《Linear and Multilinear Algebra》2013,61(2):125-131
After recalling the definition and some basic properties of finite hypergroups—a notion introduced in a recent paper by one of the authors—several non-trivial examples of such hypergroups are constructed. The examples typically consist of n n×n matrices, each of which is an appropriate polynomial in a certain tri-diagonal matrix. The crucial result required in the construction is the following: ‘let A be the matrix with ones on the super-and sub-diagonals, and with main diagonal given by a 1…a n which are non-negative integers that form either a non-decreasing or a symmetric unimodal sequence; then Ak =Pk (A) is a non-negative matrix, where pk denotes the characteristic polynomial of the top k× k principal submatrix of A, for k=1,…,n. The matrices Ak as well as the eigenvalues of A, are explicitly described in some special cases, such as (i) ai =0 for all ior (ii) ai =0 for i<n and an =1. Characters ot finite abelian hypergroups are defined, and that naturally leads to harmonic analysis on such hypergroups. 相似文献
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Richard Sinkhorn 《Linear algebra and its applications》1977,16(1):79-82
Let A be an n×n doubly stochastic matrix and suppose that 1?m?n?1. Let τ1,…,τm be m mutually disjoint zero diagonals in A, and suppose that every diagonal of A disjoint from τ1,…,τm has a constant sum. Then aall entries of A off the m zero diagonals have the value (n?m)?1. This verifies a conjecture of E.T. Wang. 相似文献
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In this paper we discuss a combinatorial problem involving graphs and matrices. Our problem is a matrix analogue of the classical problem of finding a system of distinct representatives (transversal) of a family of sets and relates closely to an extremal problem involving 1-factors and a long standing conjecture in the dimension theory of partially ordered sets. For an integer n ?1, let n denote the n element set {1,2,3,…, n}. Then let A be a k×t matrix. We say that A satisfies property P(n, k) when the following condition is satisfied: For every k-taple (x1,x2,…,xk?nk there exist k distinct integers j1,j2,…,jk so that xi= aii for i= 1,2,…,k. The minimum value of t for which there exists a k × t matrix A satisfying property P(n,k) is denoted by f(n,k). For each k?1 and n sufficiently large, we give an explicit formula for f(n, k): for each n?1 and k sufficiently large, we use probabilistic methods to provide inequalities for f(n,k). 相似文献
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Kh. D. Ikramov 《Computational Mathematics and Mathematical Physics》2009,49(5):743-747
Let s 1, ..., s n be arbitrary complex scalars. It is required to construct an n × n normal matrix A such that s i is an eigenvalue of the leading principal submatrix A i , i = 1, 2, ..., n. It is shown that, along with the obvious diagonal solution diag(s 1, ..., s n ), this problem always admits a much more interesting nondiagonal solution A. As a rule, this solution is a dense matrix; with the diagonal solution, it shares the property that each submatrix A i is itself a normal matrix, which implies interesting connections between the spectra of the neighboring submatrices A i and A i + 1. 相似文献
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A Skolem sequence of order n is a sequence S = (s1, s2…, s2n) of 2n integers satisfying the following conditions: (1) for every k ∈ {1, 2,… n} there exist exactly two elements si,Sj such that Si = Sj = k; (2) If si = sj = k,i < j then j ? i = k. In this article we show the existence of disjoint Skolem, disjoint hooked Skolem, and disjoint near-Skolem sequences. Then we apply these concepts to the existence problems of disjoint cyclic Steiner and Mendelsohn triple systems and the existence of disjoint 1-covering designs. © 1993 John Wiley & Sons, Inc. 相似文献
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Stephen L. Lipscomb 《Semigroup Forum》1992,45(1):249-260
One presentation of the alternating groupA n hasn?2 generatorss 1,…,sn?2 and relationss 1 3 =s i 2 =(s1?1si)3=(sjsk)2=1, wherei>1 and |j?k|>1. Against this backdrop, a presentation of the alternating semigroupA n c )A n is introduced: It hasn?1 generatorss 1,…,S n?2,e, theA n-relations (above), and relationse 2=e, (es 1)4, (es j)2=(es j)4,es i=s i s 1 -1 es 1, wherej>1 andi≥1. 相似文献
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Barbara L. Osofsky 《代数通讯》2013,41(7):2071-2080
An = An(?) denotes the unique left distributive binary system on {0, 1,…,2n?1) that satisfies a ? 1 = a + 1 mod 2nfor all a ? An, and on(a) = k indicates the period 2kof a ? An (if b,c ? An, then a ? b = a ? c iff b ‵ c mod 2kand if 0 < b < c < 2kthen a < a ? b < a ? c). Amongs others, we prove that ot2(2t?2) ≤ t holds for every integer t ≥ 2, the equality taking place iff t is of the form 22? for an integer s ≥ 0. 相似文献
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John Goldwasser 《Linear and Multilinear Algebra》2013,61(3-4):185-188
Let Ωn be the set of all n × n doubly stochastic matrices, let Jn be the n × n matrix all of whose entries are 1/n and let σ k (A) denote the sum of the permanent of all k × k submatrices of A. It has been conjectured that if A ε Ω n and A ≠ JJ then gA,k (θ) ? σ k ((1 θ)Jn 1 θA) is strictly increasing on [0,1] for k = 2,3,…,n. We show that if A = A 1 ⊕ ⊕At (t ≥ 2) is an n × n matrix where Ai for i = 1,2, …,t, and if for each i gAi,ki (θ) is non-decreasing on [0.1] for kt = 2,3,…,ni , then gA,k (θ) is strictly increasing on [0,1] for k = 2,3,…,n. 相似文献
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Let = (1, 2, …, n)′ be the least-squares estimator of θ = (θ1, θ2, …, θn)′ by the realization of the process y(t) = Σk = 1nθkfk(t) + ξ(t) on the interval T = [a, b] with f = (f1, f2, …, fn)′ belonging to a certain set X. The process satisfies E(ξ(t))≡0 and has known continuous covariance r(s, t) = E(ξ(s)ξ(t)) on T × T. In this paper, A-, D-, and Ds-optimality are used as criteria for choosing f in X. A-, D-, and Ds-optimal models can be constructed explicitly by means of r. 相似文献