The Dirichlet integral provides a formula for the volume over the k-dimensional simplex ω={x1,…,xk: xi?0, i=1,…,k, s?∑k1xi?T}. This integral was extended by Liouville. The present paper provides a matrix analog where now the region becomes , where now each Vi is a p×p symmetric matrix and A?B means that A?B is positive semidefinite. 相似文献
It has been conjectured that if A is a doubly stochastic n>× n matrix such that per A(i, j)≥perA for all i, j, then either A = Jn, then n × n matrix with each entry equal to , or, up to permutations of rows and columns, , where Pn is a full cycle permutation matrix. This conjecture is proved. 相似文献
Let (m?n) denote the linear space of all m × n complex or real matrices according as = or . Let c=(c1,…,cm)≠0 be such that c1???cm?0. The c-spectral norm of a matrix A?m×n is the quantity . where σ1(A)???σm(A) are the singular values of A. Let d=(d1,…,dm)≠0, where d1???dm?0. We consider the linear isometries between the normed spaces and , and prove that they are dual transformations of the linear operators which map (d) onto (c), where . 相似文献
Suppose A, D1,…,Dm are n × n matrices where A is self-adjoint, and let . It is shown that if , then the spectrum of X is majorized by the spectrum of A. In general, without assuming any condition on D1,…,Dm, a result is obtained in terms of weak majorization. If each Dk is a diagonal matrix, then X is equal to the Schur (entrywise) product of A with a positive semidefinite matrix. Thus the results are applicable to spectra of Schur products of positive semidefinite matrices. If A, B are self-adjoint with B positive semidefinite and if bii = 1 for each i, it follows that the spectrum of the Schur product of A and B is majorized by that of A. A stronger version of a conjecture due to Marshall and Olkin is also proved. 相似文献
Let A be an n-square normal matrix over , and Qm, n be the set of strictly increasing integer sequences of length m chosen from 1,…, n. For α,β∈Qm, n denote by A[α|β] the submatrix obtained from A by using rows numbered α and columns numbered β. For k∈{0,1,…,m} write z.sfnc;α∩β|=k if there exists a rearrangement of 1,…,m, say i1,…,ik, ik+1,…,im, such that α(ij)=β(ij), j=1,…,k, and {α(ik+1),…,α(im)};∩{β(ik+1),…,β(im)}=ø. Let be the group of n-square unitary matrices. Define the nonnegative number , where |α∩β|=k. Theorem 1 establishes a bound for ?k(A), 0?k<m?1, in terms of a classical variational inequality due to Fermat. Let A be positive semidefinite Hermitian, n?2m. Theorem 2 leads to an interlacing inequality which, in the case n=4, m=2, resolves in the affirmative the conjecture that . 相似文献
Let be a polynomial in the variables x1,…,xp with nonnegative real coefficients which sum to one, let A1,…,Ap be stochastic matrices, and let be the stochastic matrix which is obtained from ? by substituting the Kronecker product of An11,…,Anppfor each term Xn11·?·Xnpp. In this paper, we present necessary and sufficient conditions for the Cesàro limit of the sequence of the powers of to be equal to the Kronecker product of the Cesàro limits associated with each of A1,…,Ap. These conditions show that the equality of these two matrices depends only on the number of ergodic sets under and?or the cyclic structure of the ergodic sets under A1,…,Ap, respectively. As a special case of these results, we obtain necessary and sufficient conditions for the interchangeability of the Kronecker product and the Cesàro limit operator. 相似文献
Let Ω be a finite set with k elements and for each integer let (n-tuple) and and aj ≠ aj+1 for some 1 ≦ j ≦ n ? 1}. Let {Ym} be a sequence of independent and identically distributed random variables such that P(Y1 = a) = k?1 for all a in Ω. In this paper, we obtain some very surprising and interesting results about the first occurrence of elements in and in Ω?n with respect to the stochastic process {Ym}. The results here provide us with a better and deeper understanding of the fair coin-tossing (k-sided) process. 相似文献
Let a complex pxn matrix A be partitioned as A′=(A′1,A′2,…,A′k). Denote by ?(A), A′, and A? respectively the rank of A, the transpose of A, and an inner inverse (or a g-inverse) of A. Let A(14) be an inner inverse of A such that A(14)A is a Hermitian matrix. Let B=(A(14)1,A(14)2,…,Ak(14)) and .Then the product of nonzero eigenvalues of BA (or AB) cannot exceed one, and the product of nonzero eigenvalues of BA is equal to one if and only if either B=A(14) or for all i ≠ j,i, j=1,2,…,k . The results of Lavoie (1980) and Styan (1981) are obtained as particular cases. A result is obtained for k=2 when the condition is no longer true. 相似文献
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
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
Results on partition of energy and on energy decay are derived for solutions of the Cauchy problem . Here the Aj's are constant, k × k Hermitian matrices, x = (x1,…, xn), t represents time, and u = u(t, x) is a k-vector. It is shown that the energy of Mu approaches a limit , where M is an arbitrary matrix; that there exists a sufficiently large subspace of data ?, which is invariant under the solution group U0(t) and such that depending on ? and that the local energy of nonstatic solutions decays as . More refined results on energy decay are also given and the existence of wave operators is established, considering a perturbed equation at infinity. 相似文献
If θ is a norm on Cn, then the mapping from Mn(C) (=Cn × n) into R is called the logarithmic derivative induced by the vector norm θ. In this paper we generalize this concept to a mapping γ from Mn(C) into Mk(R), where k ? n. Denoting by α(B) the spectral abscissa of a square matrix B (the largest of the real parts of the eigenvalues), we show, in particular, that α(A) ?α(γ(A)). As a byproduct we obtain simple sufficient conditions for the stability of a matrix. 相似文献
Let θ(k, p) be the least s such that the congruence has a nontrivial solution. Let θ(k) = {max θ(k, p)| p > 1 + 2k}. The purpose of this note is to prove the following conjecture of S. Chowla: . 相似文献
Let and denote respectively the space of n×n complex matrices and the real space of n×n hermitian matrices. Let p,q,n be positive integers such that p?q?n. For , the (p,q)-numerical range of A is the set , where Cp(X) is the pth compound matrix of X, and Jq is the matrix Iq?On-q. Let denote n or . The problem of determining all linear operators T: → such that is treated in this paper. 相似文献
n independent adiabatic invariants in involution are found for a slowly varying Hamiltonian system of order 2n × 2n. The Hamiltonian system considered is , where A(t) is a 2n × 2n real matrix with distinct, pure imaginary eigen values for each t? [?∞, ∞], and , for all j > 0. The adiabatic invariants Is(u, t), s = 1,…, n are expressed in terms of the eigen vectors of A(t). Approximate solutions for the system to arbitrary order of ? are obtained uniformly for t? [?∞, ∞]. 相似文献
The permanent function is used to determine geometrical properties of the set of all n × n nonnegative doubly stochastic matrices. If is a face of , then corresponds to an n × n (0, 1)-matrix A, where the permanent of A is the number of vertices of . If A is fully indecomposable, then the dimension of equals σ(A) ? 2n + 1, where σ(A) is the number of 1's in A. The only two-dimensional faces of are triangles and rectangles. For n ? 6, has four types of three-dimensional faces. The facets of the faces of are characterized. Faces of which are simplices are determined. If is a face of which is two-neighborly but not a simplex, then has dimension 4 and six vertices. All k-dimensional faces with k + 2 vertices are determined. The maximum number of vertices of a k-dimensional face is 2k. All k-dimensional faces with at least 2k?1 + 1 vertices are determined. 相似文献
Let Xn,1 ≤ Xn,2 ≤ … ≤ Xn,n be the ordered variables corresponding to a random sample of size n with respect to a family of probability measures {Pθ:θ ∈ Θ} where Θ is an open subset of the real line. In many practical situations the Xn,i are the observables and experimentation must be curtailed prior to Xn,n. If τn is a stopping variable adapted to the σ-fields {σ(Xn,1,…,Xn,k): 1 ≤ k ≤ n} and Pn,θ the projection of Pθ onto σ(Xn,1,…,Xn,τn), the local asymptotic normality of the stopped progressively censored likelihood ratio statistics is established with θ, and θ, u held fixed, under certain conditions on the underlying distribution and on τn. Conditions are also given to ensure the local asymptotic normality of likelihood ratio statistics where the underlying observations are given in a series scheme. 相似文献
The following results are proved: Let A = (aij) be an n × n complex matrix, n ? 2, and let k be a fixed integer, 1 ? k ? n ? 1.(1) If there exists a monotonic G-function f = (f1,…,fn) such that for every subset of S of {1,…,n} consisting of k + 1 elements we have then the rank of A is ? n ? k + 1. (2) If A is irreducible and if there exists a G-function f = (f1,…,fn) such that for every subset of S of {1,…,n} consisting of k + 1 elements we have then the rank of A is ? n ? k + 1 if k ? 2, n ? 3; it is ? n ? 1 if k = 1. 相似文献
For 1 ? p ? ∞, let , be the lp norm of an m × n complex A = (αij) ?Cm × n. The main purpose of this paper is to find, for any p, q ? 1, the best (smallest) possible constants τ(m, k, n, p, q) and σ(m, k, n, p, q) for which inequalities of the form hold for all A?Cm × k, B?Ck × n. This leads to upper bounds for inner products on Ck and for ordinary lp operator norms on Cm × n. 相似文献
Let be a real or complex n × n interval matrix. Then it is shown that the Neumann series is convergent iff the sequence {k} converges to the null matrix , i.e., iff the spectral radius of the real comparison matrix constructed in [2] is less than one. 相似文献