共查询到20条相似文献,搜索用时 31 毫秒
1.
David L Johnson 《Journal of Mathematical Analysis and Applications》1982,89(2):359-369
It is shown that the set m × n of complex m × n matrices forms a lower semilattice under the partial ordering A ? B defined by denotes the conjugate transpose of A. As a special case of a result for division rings, it is further shown that, over any field F, form = n = 2 and any proper involution 1 of F2 × 2, the corresponding intersections A ∩ B all exist. 相似文献
2.
The Schur product of two n×n complex matrices A=(aij), B=(bij) is defined by A°B=(aijbij. By a result of Schur [2], the algebra of n×n matrices with Schur product and the usual addition is a commutative Banach algebra under the operator norm (the norm of the operator defined on n by the matrix). For a fixed matrix A, the norm of the operator B?A°B on this Banach algebra is called the Schur multiplier norm of A, and is denoted by ∥A∥m. It is proved here that for all unitary U (where ∥·∥ denotes the operator norm) iff A is a scalar multiple of a unitary matrix; and that ∥A∥m=∥A∥ iff there exist two permutations P, Q, a p×p (1?p?n) unitary U, an (n?p)×(n?p)1 contraction C, and a nonnegative number λ such that and this is so iff , where ā is the matrix obtained by taking entrywise conjugates of A. 相似文献
3.
4.
Zeev Schuss 《Journal of Mathematical Analysis and Applications》1977,59(2):227-241
Let A and B be uniformly elliptic operators of orders 2m and 2n, respectively, m > n. We consider the Dirichlet problems for the equations (?2(m ? n)A + B + λ2nI)u? = f and (B + λ2nI)u = f in a bounded domain Ω in Rk with a smooth boundary ?Ω. The estimate is derived. This result extends the results of [7, 9, 10, 12, 14, 15, 18]by giving estimates up to the boundary, improving the rate of convergence in ?, using lower norms, and considering operators of higher order with variable coefficients. An application to a parabolic boundary value problem is given. 相似文献
5.
Motivated by models from stochastic population biology and statistical mechanics, we proved new inequalities of the form , where A and B are n × n complex matrices, 1<n<∞, and ? is a real-valued continuous function of the eigenvalues of its matrix argument. For example, if A is essentially nonnegative, B is diagonal real, and ? is the spectral radius, then (1) holds; if in addition A is irreducible and B has at least two different diagonal elements, then the inequality (1) is strict. The proof uses Kingman's theorem on the log-convexity of the spectral radius, Lie's product formula, and perturbation theory. We conclude with conjectures. 相似文献
6.
Thomas H. Pate 《Linear algebra and its applications》1976,14(3):285-292
Suppose each of m, n, and k is a positive integer, k ? n, A is a (real-valued) symmetric n-linear function on Em, and B is a k-linear symmetric function on Em. The tensor and symmetric products of A and B are denoted, respectively, by A ?B and A?B. The identity is proven by Neuberger in [1]. An immediate consequence of this identity is the inequality In this paper a necessary and sufficient condition for is given. It is also shown that under certain conditions the inequality can be considerably improved. This improvement results from an analysis of the terms 6A?qB6, 1?q?n, appearing in the identity. 相似文献
7.
Ryuzaburo Noda 《Journal of Combinatorial Theory, Series A》1978,24(1):89-95
t?(2k, k, λ) designs having a property similar to that of Hadamard 3-designs are studied. We consider conditions (i), (ii), or (iii) for t?(2k, k, λ) designs: (i) The complement of each block is a block. (ii) If A and B are a complementary pair of blocks, then ∥ A ∩ C ∥ = ∥ B ∩ C ∥ ± u holds for any block C distinct from A and B, where u is a positive integer. (iii) if A and B are a complementary pair of blocks, then ∥ A ∩ C ∥ = ∥ B ∩ C ∥ or ∥ A ∩ C ∥ = ∥ B ∩ C ∥ ± u holds for any block C distinct from A and B, where u is a positive integer. We show that a t?(2k, k, λ) design with t ? 2 and with properties (i) and (ii) is a 3?(2u(2u + 1), u(2u + 1), u(2u2 + u ? 2)) design, and that a t?(2k, k, λ) design with t ? 4 and with properties (i) and (iii) is the 5-(12, 6, 1) design, the 4-(8, 4, 1) design, a design, or a design. 相似文献
8.
T Ando 《Journal of Mathematical Analysis and Applications》1979,70(1):72-84
When A and B are n × n positive semi-definite matrices, and C is an n × n Hermitian matrix, the validity of a quadratic inequality is shown to be equivalent to the existence of an n × n unitary matrix W such that Some related inequalities are also discussed. 相似文献
9.
V.B Headley 《Journal of Mathematical Analysis and Applications》1985,108(1):283-292
Let D(?) be the Doob's class containing all functions f(z) analytic in the unit disk Δ such that f(0) = 0 and lim on an arc A of ?Δ with length . It is first proved that if f?D(?) then the spherical norm ∥ f ∥ = supz?Δ, where C1 = limn→∞. Next, U represents the Seidel's class containing all non-constant functions f(z) bounded analytic in Δ such that almost everywhere. It is proved that inff?U∥f∥ = 0, and if f has either no singularities or only isolated singularities on ?Δ, then ∥f∥ ? C1. Finally, it is proved that if f is a function normal in Δ, namely, the norm ∥f∥< ∞, then we have the sharp estimate ∥fp∥ ? p∥f∥, for any positive integer p. 相似文献
10.
Yueh-er Kuo 《Journal of Mathematical Analysis and Applications》1976,56(2):346-350
Let be column-wise partitioned matrices over complex numbers. Then an extended Kronecker product is , where Ai ? Bi is the Kronecker product of Ai and Bi. Some properties of an extended Kronecker product of matrices are investigated. The properties of the solutions of the systems of linear equations whose coefficient matrices are extended Kronecker products of matrices are studied. 相似文献
12.
Yik-Hoi Au-Yeung 《Linear algebra and its applications》1975,10(1):71-76
Let A and B be two n×n real symmetric matrices. A theorem of Calabi and Greub-Milnor states that if n?3 and A and B satisfy the condition for all nonzero vectors u, then there is a linear combination of A and B that is definite. In this note, the author proves two theorems of the semi-definiteness of a nontrivial linear combination of A and B by replacing the condition (1) by another condition. One of these theorems is a generalization of the theorem of Greub-Milnor and Calabi. 相似文献
13.
If f is a positive function on (0, ∞) which is monotone of order n for every n in the sense of Löwner and if Φ1 and Φ2 are concave maps among positive definite matrices, then the following map involving tensor products: is proved to be concave. If Φ1 is affine, it is proved without use of positivity that the map is convex. These yield the concavity of the map (0<p?1) (Lieb's theorem) and the convexity of the map (0<p?1), as well as the convexity of the map .These concavity and convexity theorems are then applied to obtain unusual estimates, from above and below, for Hadamard products of positive definite matrices. 相似文献
14.
Juan C. Peral 《Journal of Functional Analysis》1980,36(1):114-145
Let u(x, t) be the solution of utt ? Δxu = 0 with initial conditions . Consider the linear operator . (Here g = 0.) We prove for t fixed the following result. Theorem 1: T is bounded in Lp if and only if . Theorem 2: If the coefficients are variables in C and constant outside of some compact set we get: (a) If n = 2k the result holds for . (b) If n = 2k ? 1, the result is valid for . This result are sharp in the sense that for p such that we prove the existence of in such a way that . Several applications are given, one of them is to the study of the Klein-Gordon equation, the other to the completion of the study of the family of multipliers and finally we get that the convolution against the kernel is bounded in H1. 相似文献
15.
Let Ω = {1, 0} and for each integer n ≥ 1 let (n-tuple) and for all k = 0,1,…,n. Let {Ym}m≥1 be a sequence of i.i.d. random variables such that . For each A in , let TA be the first occurrence time of A with respect to the stochastic process {Ym}m≥1. R. Chen and A.Zame (1979, J. Multivariate Anal. 9, 150–157) prove that if n ≥ 3, then for each element A in , there is an element B in such that the probability that TB is less than TA is greater than . This result is sharpened as follows: (I) for n ≥ 4 and 1 ≤ k ≤ n ? 1, each element A in , there is an element B also in such that the probability that TB is less than TA is greater than ; (II) for n ≥ 4 and 1 ≤ k ≤ n ? 1, each element A = (a1, a2,…,an) in , there is an element C also in such that the probability that TA is less than TC is greater than if n ≠ 2m or n = 2m but ai = ai + 1 for some 1 ≤ i ≤ n?1. These new results provide us with a better and deeper understanding of the fair coin tossing process. 相似文献
16.
For a given pair such that A is cyclic and b is a cyclic generator (with respect to A) of , it is shown that for every nonnegative integer m we can find a nonnegative integer t and a sequence ,so that a the zeros of the rational function det P(z), where f, lie in the open unit disc in the complex plane. The result is directly applicable to a stabilizability problem for linear systems with a time delay in control action. 相似文献
17.
Pascal Cherrier 《Journal of Functional Analysis》1984,57(2):154-206
Nonlinear Neumann problems on riemannian manifolds. Let () be a C∞ compact riemannian manifold of dimension n ? 2 whose boundary B is an (n ? 1)-dimensional submanifold and let be the interior of . Study of Neumann problems of the form: , where, for every are bounded by . Application to the determination of a conformal metric for which the scalar curvature of M and the mean curvature of B take prescribed values. 相似文献
18.
Man Kam Kwong 《Linear algebra and its applications》1977,18(3):223-228
Let A and B be two n×n non-negative matrices. We write A ? B iff for all column vectors u in Cn. Here u1 is the conjugate transpose of u. In this paper are stated equivalent conditions under which Ak ? Bk for all natural numbers k. The result is then generalized to Hermitian operators in a Hilbert space. 相似文献
19.
Let A be a real symmetric n × n matrix of rank k, and suppose that A = BB′ for some real n × m matrix B with nonnegative entries (for some m). (Such an A is called completely positive.) It is shown that such a B exists with , where 2N is the maximal number of (off-diagonal) entries which equal zero in a nonsingular principal submatrix of A. An example is given where the least m which works is (k odd), (k even). 相似文献
20.
Rudolf Wegmann 《Journal of Mathematical Analysis and Applications》1976,56(1):113-132
For an n × n Hermitean matrix A with eigenvalues λ1, …, λn the eigenvalue-distribution is defined by · number {λi: λi ? x} for all real x. Let An for n = 1, 2, … be an n × n matrix, whose entries aik are for i, k = 1, …, n independent complex random variables on a probability space (Ω, , p) with the same distribution Fa. Suppose that all moments | a | k, k = 1, 2, … are finite, a=0 and | a | 2. Let with complex numbers θσ and finite products Pσ of factors A and (= Hermitean conjugate) be a function which assigns to each matrix A an Hermitean matrix M(A). The following limit theorem is proved: There exists a distribution function G0(x) = G1x) + G2(x), where G1 is a step function and G2 is absolutely continuous, such that with probability converges to G0(x) as n → ∞ for all continuity points x of G0. The density g of G2 vanishes outside a finite interval. There are only finitely many jumps of G1. Both, G1 and G2, can explicitly be expressed by means of a certain algebraic function f, which is determined by equations, which can easily be derived from the special form of M(A). This result is analogous to Wigner's semicircle theorem for symmetric random matrices (E. P. Wigner, Random matrices in physics, SIAM Review9 (1967), 1–23). The examples , , , r = 1, 2, …, are discussed in more detail. Some inequalities for random matrices are derived. It turns out that with probability 1 the sharpened form of Schur's inequality for the eigenvalues λi(n) of An holds. Consequently random matrices do not tend to be normal matrices for large n. 相似文献