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1.
It is shown that if A and B are n × n complex matrices with A = A1and ∥AB ? BA∥</ 2?2(n ? 1), then there exist n × n matrices A′ and B′ with A′ = A′1such that A′B′ = B′A′ and ∥A ? A′∥? ?, ∥B ? B′∥? ?.  相似文献   

2.
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
6A · B62=q=0n(nk)(n+kk)6A?qB62
is proven by Neuberger in [1]. An immediate consequence of this identity is the inequality
6A · B 62?n+kn?16A · B 62
In this paper a necessary and sufficient condition for
6A · B 62=n+kn?6A · B 62
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   

3.
Let Ω = {1, 0} and for each integer n ≥ 1 let Ωn = Ω × Ω × … × Ω (n-tuple) and Ωnk = {(a1, a2, …, an)|(a1, a2, … , an) ? Ωnand Σi=1nai = k} for all k = 0,1,…,n. Let {Ym}m≥1 be a sequence of i.i.d. random variables such that P(Y1 = 0) = P(Y1 = 1) = 12. For each A in Ωn, 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 Ωn, there is an element B in Ωn such that the probability that TB is less than TA is greater than 12. This result is sharpened as follows: (I) for n ≥ 4 and 1 ≤ kn ? 1, each element A in Ωnk, there is an element B also in Ωnk such that the probability that TB is less than TA is greater than 12; (II) for n ≥ 4 and 1 ≤ kn ? 1, each element A = (a1, a2,…,an) in Ωnk, there is an element C also in Ωnk such that the probability that TA is less than TC is greater than 12 if n ≠ 2m or n = 2m but ai = ai + 1 for some 1 ≤ in?1. These new results provide us with a better and deeper understanding of the fair coin tossing process.  相似文献   

4.
Suppose A, D1,…,Dm are n × n matrices where A is self-adjoint, and let X = Σmk = 1DkAD1k. It is shown that if ΣDkD1k = ΣD1kDk = I, 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   

5.
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 Cn 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 ∥Am. It is proved here that ∥A∥=∥U1AU∥m for all unitary U (where ∥·∥ denotes the operator norm) iff A is a scalar multiple of a unitary matrix; and that ∥Am=∥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
A=λPU00CQ;
and this is so iff ∥A°A?∥=∥A∥2, where ā is the matrix obtained by taking entrywise conjugates of A.  相似文献   

6.
For any prime p, the sequence of Bell exponential numbers Bn is shown to have p ? 1 consecutive values congruent to zero (mod p), beginning with Bm, where m ≡ 1 ? (pp ? 1)(p ? 1)2 (mod(pp ? 1)(p ? 1)). This is an improvement over previous results on the maximal strings of zero residues of the Bell numbers. Similar results are obtained for the sequence of generalized Bell numbers An generated by e?(ex ? 1) = Σn = 0 Anxnn!.  相似文献   

7.
It is shown, for n ? m ? 1, that there exist inner maps Φ: BnBm with boundary values Φ1: Bn → Bm such that σm(A) = σn1?1(A)). where σn and σm are the Haar measures on ?Bn and ?Bm, respectively, and A ? Bn is an arbitrary Borel set.  相似文献   

8.
For a given pair (A,b)∈Rn×n×Rn×1 such that A is cyclic and b is a cyclic generator (with respect to A) of Rn×1, it is shown that for every nonnegative integer m we can find a nonnegative integer t and a sequence {fj}tj=0,fjR1×n,so that a the zeros of the rational function det P(z), where P(z) = zI ? A ? ∑tj=0z-(m+j)b?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   

9.
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 m?12k(k+1)?N, 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+1)24 (k odd),k(k+2)4 (k even).  相似文献   

10.
Let A be a real or complex n × n interval matrix. Then it is shown that the Neumann series Σk=0Ak is convergent iff the sequence {Ak} converges to the null matrix O, i.e., iff the spectral radius of the real comparison matrix B constructed in [2] is less than one.  相似文献   

11.
For a sequence A = {Ak} of finite subsets of N we introduce: δ(A) = infm?nA(m)2n, d(A) = lim infn→∞ A(n)2n, where A(m) is the number of subsets Ak ? {1, 2, …, m}.The collection of all subsets of {1, …, n} together with the operation a ∪ b, (a ∩ b), (a 1 b = a ∪ b ? a ∩ b) constitutes a finite semi-group N (semi-group N) (group N1). For N, N we prove analogues of the Erdös-Landau theorem: δ(A+B) ? δ(A)(1+(2λ)?1(1?δ(A>))), where B is a base of N of the average order λ. We prove for N, N, N1 analogues of Schnirelmann's theorem (that δ(A) + δ(B) > 1 implies δ(A + B) = 1) and the inequalities λ ? 2h, where h is the order of the base.We introduce the concept of divisibility of subsets: a|b if b is a continuation of a. We prove an analog of the Davenport-Erdös theorem: if d(A) > 0, then there exists an infinite sequence {Akr}, where Akr | Akr+1 for r = 1, 2, …. In Section 6 we consider for N∪, N∩, N1 analogues of Rohrbach inequality: 2n ? g(n) ? 2n, where g(n) = min k over the subsets {a1 < … < ak} ? {0, 1, 2, …, n}, such that every m? {0, 1, 2, …, n} can be expressed as m = ai + aj.Pour une série A = {Ak} de sous-ensembles finis de N on introduit les densités: δ(A) = infm?nA(m)2m, d(A) = lim infn→∞ A(n)2nA(m) est le nombre d'ensembles Ak ? {1, 2, …, m}. L'ensemble de toutes les parties de {1, 2, …, n} devient, pour les opérations a ∪ b, a ∩ b, a 1 b = a ∪ b ? a ∩ b, un semi-groupe fini N, N ou un groupe N1 respectivement. Pour N, N on démontre l'analogue du théorème de Erdös-Landau: δ(A + B) ? δ(A)(1 + (2λ)?1(1?δ(A))), où B est une base de N d'ordre moyen λ. On démontre pour N, N, N1 l'analogue du théorème de Schnirelmann (si δ(A) + δ(B) > 1, alors δ(A + B) = 1) et les inégalités λ ? 2h, où h est l'ordre de base. On introduit le rapport de divisibilité des enembles: a|b, si b est une continuation de a. On démontre l'analogue du théorème de Davenport-Erdös: si d(A) > 0, alors il existe une sous-série infinie {Akr}, où Akr|Akr+1, pour r = 1, 2, … . Dans le Paragraphe 6 on envisage pour N, N, N1 les analogues de l'inégalité de Rohrbach: 2n ? g(n) ? 2n, où g(n) = min k pour les ensembles {a1 < … < ak} ? {0, 1, 2, …, n} tels que pour tout m? {0, 1, 2, …, n} on a m = ai + aj.  相似文献   

12.
Let the n × n complex matrix A have complex eigenvalues λ12,…λn. Upper and lower bounds for Σ(Reλi)2 are obtained, extending similar bounds for Σ|λi|2 obtained by Eberlein (1965), Henrici (1962), and Kress, de Vries, and Wegmann (1974). These bounds involve the traces of A1A, B2, C2, and D2, where B=12 (A + A1), C=12 (A ? A1) /i, and D = AA1 ? A1A, and strengthen some of the results in our earlier paper “Bounds for eigenvalues using traces” in Linear Algebra and Appl. [12].  相似文献   

13.
14.
For an n × n Hermitean matrix A with eigenvalues λ1, …, λn the eigenvalue-distribution is defined by G(x, A) := 1n · 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, R, p) with the same distribution Fa. Suppose that all moments E | a | k, k = 1, 2, … are finite, Ea=0 and E | a | 2. Let
M(A)=σ=1s θσPσ(A,A1)
with complex numbers θσ and finite products Pσ of factors A and A1 (= 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 1 G(x, M(Ann12)) 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 ArA1r, Ar + A1r, ArA1r ± A1rAr, 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
lim supn→∞i=1ni(n)|2?6An62? 0.8228…
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   

15.
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 (x1Ax)12(x1Bx)12 ? ¦x1Cx¦ is shown to be equivalent to the existence of an n × n unitary matrix W such that A12WB12 + B12W1A12 = 2C. Some related inequalities are also discussed.  相似文献   

16.
We show that any m × n matrix A, over any field, can be written as a product, LSP, of three matrices, where L is a lower triangular matrix with l's on the main diagonal, S is an m × n matrix which reduces to an upper triangular matrix with nonzero diagonal elements when the zero rows are deleted, and P is an n × n permutation matrix. Moreover, L, S, and P can be found in O(mα?1n) time, where the complexity of matrix multiplication is O(mα). We use the LSP decomposition to construct fast algorithms for some important matrix problems. In particular, we develop O(mα?1n) algorithms for the following problems, where A is any m × n matrix: (1) Determine if the system of equations Ax = b (where b is a column vector) has a solution, and if so, find one such solution. (2) Find a generalized inverse, A1, of A (i.e., AA1A = A). (3) Find simultaneously a maximal independent set of rows and a maximal independent set of columns of A.  相似文献   

17.
Let Fm×n (m?n) denote the linear space of all m × n complex or real matrices according as F=C or R. Let c=(c1,…,cm)≠0 be such that c1???cm?0. The c-spectral norm of a matrix A?Fm×n is the quantity
6A6ci=Imciσi(A)
. 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 (Fn,∥·∥c) and (Fn,∥·∥d), and prove that they are dual transformations of the linear operators which map L(d) onto L(c), where
L(c)= {X?Fm×n:X has singular values c1,…,cm}
.  相似文献   

18.
If we change the sign of p ? m columns (or rows) of an m × m positive definite symmetric matrix A, the resultant matrix B has p negative eigenvalues. We give systems of inequalities for the eigenvalues of B and of the matrix obtained from B by deleting one row and column. To obtain these, we first develop characterizations of the eigenvalues of B which are analogous to the minimum-maximum properties of the eigenvalues of a symmetric A, i.e. the Courant-Fischer theorem. These results arose from studying probability distributions on the hyperboloid of revolution
x21 + ? + x2m?p ? x2m ? p + 1 ? ? ? x2m = 1
. By contrast, the familiar results are associated with the sphere x21 + ? + x2m = 1.  相似文献   

19.
Assume that A is an m × n real matrix with m?n and rank(A)=n. Let b be a real vector with m components and consider the problem of finding x=A2b, where A2=(ATA)?1AT. A general scheme for solving this problem is described. Many well-known and commonly used in practice direct methods can be found as special cases within the general scheme. This is illustrated by four examples. The general scheme can be used to study the common properties of the direct methods. The usefulness of this approach in the efforts to improve the efficiency of the direct methods for sparse matrices is demonstrated by formulating an algorithm that can be applied to any particular method belonging to the general scheme. This algorithm is implemented in several subroutines solving linear algebraic problems by different direct methods. Numerical results, obtained in a wide range of runs with these subroutines, are given.  相似文献   

20.
Let Fn denote the ring of n×n matrices over the finite field F=GF(q) and let A(x)=ANxN+ ?+ A1x+A0?Fn[x]. A function ?:Fn→Fn is called a right polynomial function iff there exists an A(x)?Fn[x] such that ?(B)=ANBN+?+A1B+ A0 for every B?Fn. This paper obtains unique representations for and determines the number of right polynomial functions.  相似文献   

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