线性流形上实对称半正定阵的一类反问题

1　引　　言文中记Rn×m为所有n×m阶实阵集合,SRn×n为所有n阶实对称阵集合,Pn表示所有n阶实对称半正定阵集合,A≥0表示方阵A对称半正定.A+、R(A)、N(A)分别表示矩阵A的Moore-Penrose广义逆,列空间和零空间,‖·‖表示Froblnius范数.对于Z.Y∈Rn×k,令S={A∈Pn|AZ=Y,ZTY∈PK,R(YT)=R(YTZ)}(1.1)　　现考虑如下问题:问题　给定X.B∈Rn×m,找A∈S,使得AX=B(1.2)　　问题　给定A∈Rn×n,找A∈SE,使得‖A-A‖=infA∈SE‖A-A‖(1.3)其中SE是问题的解集合.问题与具有重要的应用背景,当Y=ZΛ,Λ=diag(λ1,λ2,…     

一类反对称正交反对称矩阵逆特征值问题

设P为一给定的对称正交矩阵,记AARnP={A∈Rn×n‖AT=-A,(PA)T=-PA}.讨论了下列问题:问题给定X∈Cn×m,Λ=diag(λ1,λ2,…,λm).求A∈AARPn使AX=XΛ.问题设A~∈Rn×n,求A*∈SE使‖A~-A*‖=infA∈SE‖A~-A‖,其中SE为问题的解集合,‖.‖表示Frobenius范数.研究了AARPn中元素的通式,给出了问题解的一般表达式,证明了问题存在唯一逼近解A*,且得到了此解的具体表达式.     

线性流形上对称正交反对称矩阵反问题的最小二乘解

设P是n阶对称正交矩阵,如果n阶矩阵A满足AT=A和(PA)T=-PA,则称A为对称正交反对称矩阵,所有n阶对称正交反对称矩阵的全体记为SARnp.令S={A∈SARnp f(A)=‖AX-B‖=m in,X,B〗∈Rn×m本文讨论了下面两个问题问题Ⅰ给定C∈Rn×p,D∈Rp×p,求A∈S使得CTAC=D问题Ⅱ已知A~∈Rn×n,求A∧∈SE使得‖A~-A∧‖=m inA∈SE‖A~-A‖其中SE是问题Ⅰ的解集合.文中给出了问题Ⅰ有解的充要条件及其通解表达式.进而,指出了集合SE非空时,问题Ⅱ存在唯一解,并给出了解的表达式,从而得到了求解A∧的数值算法.     

具有左、右特征向量及特征值约束下逆特征值问题

§1 问题的提法R~(n×m)表示所有 n×m 阶实阵集合,(A)表示矩阵 A 的列空间,A~+表示 A 的 Moore-Penrose 广义逆,P_A=AA~+表示到(A)的正交投影核子;I_n 表示 n 阶单位阵,‖·‖_F 表示 Frobenius 范数。问题Ⅰ给定X,Y∈~(n×m),Λ=diag(λ_1,λ_2,…,λ_m)∈R~(m×m),找 A∈R~(n×m),使得问题Ⅱ给定 A~*∈R~(n×n),找∈S_E,使得‖A~*-‖_F=‖A~*-A‖_F,其中 S_E是问题Ⅰ的集合。本文讨论问题Ⅰ有解的充分与必要条件,且求出 S_E的表达式,同时给出的表达式。     

Hermitian非自反矩阵的两类反问题

记J为一广义反射矩阵,HAJn×n为关于J的n阶Hermitian非自反矩阵的集合.本文考虑如下两个问题:问题Ⅰ给定X,B∈n×m,求A∈HAJn×n,使得‖AX-B‖=min.问题Ⅱ给定X∈n×m,B∈n×n,求A∈HAJn×n,使得XHAX=B.首先利用奇异值分解讨论问题Ⅰ的解的通式,然后利用广义奇异值分解得到了问题Ⅱ有解的充分必要条件和解的通式,最后给出问题Ⅰ和Ⅱ的逼近解的具体表达式.     

线性流形上亚半正定阵的一类逆特征值问题

1　引言与引理设 Rm× n表示所有 m× n实矩阵集合 ,m=n时 ,Rm× n简记为 Rm;Rm0 表示所有 m阶亚半正定阵集合 ,即 Rm0 ={ A∈Rm× m|YTAY≥ 0 , Y∈Rm× 1 } ;ORm表示 m阶正交矩阵集合 ;A+表示矩阵 A的 Moore-Penrose广义逆 ;‖·‖表示 Frobenius范数 .In 表示 n阶单位阵 ,有时令SE={ A∈ Rm× m|‖ AE -F‖ =min,E,F∈ Rm× k} ,(1 .1 )则 SE是线性流形 .文 [1 ] ,[2 ]分别研究了 SE上实对称矩阵及实对称半正定阵的逆特征值问题 ,本文将进一步研究 SE上亚半正定阵的一类逆特征值问题 ,具体叙述如下 :问题 　给定 X,B∈R…     

THE INVERSE PROBLEM FOR PART SYMMETRIC MATRICES ON A SUBSPACE

In this paper, the following two problems are considered:Problem Ⅰ. Given S∈E Rn×p,X,B 6 Rn×m, find A ∈ SRs,n such that AX = B, where SR8,n = {A∈ Rn×n|xT(A - AT) = 0, for all x ∈ R(S)}.Problem Ⅱ. Given A* ∈ Rn×n, find A ∈ SE such that ||A-A*|| = minA∈sE||A-A*||, where SE is the solution set of Problem Ⅰ.The necessary and sufficient conditions for the solvability of and the general form of the solutions of problem Ⅰ are given. For problem Ⅱ, the expression for the solution, a numerical algorithm and a numerical example are provided.     

矩阵方程AX=B的双反对称最佳逼近解

本文主要讨论下而两个问题并得到相关结果:问题Ⅰ:给定A ∈ R~(k×n),B ∈ R~(k×n),求X ∈ BASR~(n×n),使得AX=B.问题Ⅱ:给定X* ∈R~(n×n),求X使得‖X-X~*‖=minX∈S_E‖X-X~*‖,其中S_E是问题Ⅰ的解集合,‖·‖是Frobenius范数.通过对上述问题的讨论给出了问题Ⅰ解存在的充分必要条件和其解的一般表达式同时给出了问题Ⅱ的解,算法,和数值例子.     

关于径向矩阵与谱矩阵

§1.引言与记号 设A∈C~(s×n),则称 ‖A‖=‖AX‖/‖X‖ 为A的谱模(谱范数),其中‖X‖表示向量X∈C~(n×1)的Euclid范数。即当X=(x_1,…,x_n)~(?)时,‖X‖=(XX)~1/2=sum from i=1 to n(|X_1|~2)~1/2;‖AX‖为向量AX的Euclid范数。 如众周知,我们有如下结论: 引理 1[1]、设A、B∈C~(n×n),则谱模满足范数的三个条件: 1>.恒正性:‖A‖≥0且‖A‖=0 A=0; 2>.齐次性:若α∈C,则‖αA‖=|α|·‖A‖; 3>.三角不等式:‖A+B‖≤‖A‖+‖B‖。     

关于两个最小二乘解流形的逼近性

L1={X∈Rn×m|f(X)=‖XA1-B1‖2+‖CT1X-DT1‖2=min},L2={Y∈Rn×m|g(Y)=‖YA2-B2‖2+‖CT2Y-DT2‖2=min},其中A1∈Rm×k1,B1∈Rn×k1,C1∈Rn×l1,D1∈Rm×l1,A2∈Rm×k2,B2∈Rn×k2,C2∈Rn×l2,D2∈Rm×l2均为已知矩阵,本文讨论了L1,L2两个线性流形之间的逼近性,给出了d(L1,L2)=minX∈L1,Y∈L2‖X-Y‖的具体表达式.     

RECENT PROGRESS ON SPHERICAL HARMONIC APPROXIMATION MADE BY BNU RESEARCH GROUP ——In memory of Professor Sun Yongsheng

As early as in 1990, Professor Sun Yongsheng, suggested his students at Beijing Normal University to consider research problems on the unit sphere. Under his guidance and encouragement his students started the research on spherical harmonic analysis and approximation. In this paper, we incompletely introduce the main achievements in this area obtained by our group and relative researchers during recent 5 years （2001-2005）. The main topics are： convergence of Cesaro summability, a.e. and strong summability of Fourier-Laplace series; smoothness and K-functionals; Kolmogorov and linear widths.     

Boundedness for commutators of multipliers on Hardy type spaces

In this paper, we study the commutators generalized by multipliers and a BMO function. Under some assumptions, we establish its boundedness properties from certain atomic Hardy space Hb^p（R^n） into the Lebesgue space L^p with p 〈 1.     

A unified approach to certain problems of best local approximation

In this paper we study best local quasi-rational approximation and best local approximation from finite dimensional subspaces of vectorial functions of several variables. Our approach extends and unifies several problems concerning best local multi-point approximation in different norms.     

Towards Excellence in Science Chinese Academy of Sciences

<正>May 26,2014,Beijing Science is a human enterprise in the pursuit of knowledge.The scientific revolution that occurred in the 17th Century initiated the advances of modern science.The scientific knowledge system created by     

THE 8~(th) INTERNATIONAL CONGRESS ON INDUSTRIAL AND APPLIED MATHEMATICS

<正>August 10-14,2015Beijing,ChinaThe International Congress on Industrial and Applied Mathematics(ICIAM)is the premier international congress in the field of applied mathematics held every four years under the auspices of the International Council for Industrial and Applied Mathematics.From August 10 to 14,2015,mathematicians,scientists     

Bounds for the zeros of polynomials

Let P(z)=∑↓j=0↑n ajx^j be a polynomial of degree n. In this paper we prove a more general result which interalia improves upon the bounds of a class of polynomials. We also prove a result which includes some extensions and generalizations of Enestrǒm-Kakeya theorem.     

Boundedness of commutators related to Marcinkiewicz integrals on hardy type spaces

In this paper, the authors study the boundedness of the operator [μΩ, b], the commutator generated by a function b ∈ Lipβ(Rn)(0 ＜β≤ 1) and the Marcinkiewicz integrals μΩ, on the classical Hardy spaces and the Herz-type Hardy spaces in the case Ω∈ Lipα(Sn-1)(0 ＜α≤ 1).     

Generalization of the interaction between Haar approximation and polynomial operators to higher order methods

In applications it is useful to compute the local average empirical statistics on u. A very simple relation exists when of a function f（u） of an input u from the local averages are given by a Haar approximation. The question is to know if it holds for higher order approximation methods. To do so, it is necessary to use approximate product operators defined over linear approximation spaces. These products are characterized by a Strang and Fix like condition. An explicit construction of these product operators is exhibited for piecewise polynomial functions, using Hermite interpolation. The averaging relation which holds for the Haar approximation is then recovered when the product is defined by a two point Hermite interpolation.     

Jacobi polynomials used to invert the laplace transform

Given the Laplace transform F(s) of a function f(t), we develop a new algorithm to find an approximation to f(t) by the use of the classical Jacobi polynomials. The main contribution of our work is the development of a new and very effective method to determine the coefficients in the finite series expansion that approximation f(t) in terms of Jacobi polynomials. Some numerical examples are illustrated.