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…     

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。     

Hoffman-Wielandt定理的推广及应用

1　引言及记号用 Rn× n表示所有 n× n阶实矩阵的集合 ,用 Sn× n,Sn× n+及 Sn× n++分别表示所有 n×n实对称矩阵 ,实对称半正定矩阵及实对称正定矩阵的集合 ,用 Tr(M)表示矩阵 M的迹 ,对 A,B∈ Rn× n.定义其内积为 A×B=Tr(ATB) .考虑如下半正定线性互补问题 :求 X,Y∈ Sn× n使Y =L (X) +Q,X≥ O,Y≥ O,X× Y =0 ,(1)其中 Q∈ Sn× n,L :Sn× n→ Sn× n为线性算子 ,而 X≥ O表示 X∈ Sn× n+(O表示零矩阵 ) .若 L:Sn× n→Sn× n满足X× L (X)≥ 0 ,　　 X∈ Sn× n. (2 )则称其为单调算子 ,而相应的问题称为单…     

子空间上的一类矩阵反问题

1 引言 设Rn×m为所有n×m实矩阵的集合,ASRn×n为n阶实反对称矩阵的集合,ORn×n 为n阶实正交矩阵的全体. In是n阶单位矩阵,A+,R(A),N(A)分别表示矩阵A的 Moore-Penrose广义逆、值域及零空间,并记EA=I-AA+,FA=I-A+A(I为单位矩 阵,A为任意矩阵).对A=(aij),B=(bij)∈Rn×m,A*B=(aijbij)表示矩阵A与B 的Hadamard积.在Rn×m上定义矩阵A与B的内积为(A,B)=tr(BT A),则由此内积 导出的范数‖A‖=(A,A)~(1/2)是矩阵的Frobenius范数,并且Rn×m构成一个完备的内积 空间.     

线性流形上的广义中心对称矩阵反问题

设R∈Cn×n是满足R=RH=R-1≠±In的广义反射矩阵.若A∈Cn×n满足RAR=A,则称A为n阶广义中心对称矩阵,n阶广义中心对称矩阵的全体记为GCSCn×n.令X1,Z1∈Cn×k1,Y1,W1∈Cn×l1,S={A|‖AX1-Z1‖2+‖Y1HA-W1H‖2=min,A∈GCSCn×n},本文研究如下问题.问题Ⅰ.给定矩阵Z2,X2∈Cn×k2,Y2,W2∈Cn×l2,求A∈S,使得其中‖·‖是Frobenius范数.问题Ⅱ.给定矩阵A∈Cn×n,求A∈SE,使得其中SE是问题Ⅰ的解集合.本文给出了问题Ⅰ解集合SE的表达式,并导出了矩阵方程AX2=Z2,Y2HA=W2H有解A∈S的充分必要条件及其通解表达式,并给出了问题Ⅱ解的表达式以及求解问题Ⅱ的数值方法和数值例子.     

双反对称矩阵反问题的最小二乘解

1 引 言Rn×m表示所有n×m阶实矩阵集合,Rrn×m表示Rn×m中秩为r的子集;ORn×m表示所有n阶正交阵的集合;A+表示A的Moore-Penrose广义逆;Iκ表示κ阶单位阵;||·||表示Frobenius范数;ASRn×m表示n阶实反对称阵的全体;A＊B表示A与B的Hadamard乘     

矩阵方程的最小二乘解

1　引言与引理设 Rm× n表示所有 m× n阶实矩阵的集合 ,ORn× n为所有 n阶实正交矩阵的全体 ,In 是 n阶单位矩阵 .AT、A+、rank A分别表示矩阵 A的转置、MP逆及秩 ;‖·‖是矩阵的Frobenius范数 .此外 ,对于 A =(αij)∈ Rs× s,B =(βij)∈ Rs× s,A * B表示 A与 B的Hadamard积 ,其定义为 :A* B=(αijβij) 1≤ i,j≤ s,现考虑如下问题 :问题 P　给定 A∈Rn× m,B∈Rp× m,D∈Rm× m求 X∈Rn× p,使得Φ =‖ ATXB - BTXTA - D‖ =m in　　我们知道 ,矩阵方程 ATX B- BTXTA=D在自动控制理论中有很重要的作用[1 ,2 ] .…     

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.     

矩阵方程XTAX=B的一类反问题

1引言 本文用Rn×m表示所有n×m实矩阵全体;SR0n×n表示所有n阶实对称半正定矩阵全体;In表示n阶单位矩阵;A-,A+分别表示矩阵A的一个广义逆和Moore-Penrose广义逆;A≥0表示A为对称半正定矩阵;Sn=(en,en-1,…,e1)∈Rn×n,其中ei为单位阵In的第i列; [n/2]表示不超过n/2的最大整数.     

ON A THEOREM OF A.A. LYAPUNOV

Abstract

In this note we obtain some extensions and an approximation of the Lyapunov convexity theorem by means of the bilinear integration of a set-valued function. The integration is performed successively with respect to a non-atomic, a direct sum and a Darboux vector measure. The necessary counterexamples are provided.     

A Priori and A Posteriori: A Bootstrapping Relationship

The distinction between a priori and a posteriori knowledge has been the subject of an enormous amount of discussion, but the literature is biased against recognizing the intimate relationship between these forms of knowledge. For instance, it seems to be almost impossible to find a sample of pure a priori or a posteriori knowledge. In this paper, it will be suggested that distinguishing between a priori and a posteriori is more problematic than is often suggested, and that a priori and a posteriori resources are in fact used in parallel. We will define this relationship between a priori and a posteriori knowledge as the bootstrapping relationship. As we will see, this relationship gives us reasons to seek for an altogether novel definition of a priori and a posteriori knowledge. Specifically, we will have to analyse the relationship between a priori knowledge and a priori reasoning, and it will be suggested that the latter serves as a more promising starting point for the analysis of aprioricity. We will also analyse a number of examples from the natural sciences and consider the role of a priori reasoning in these examples. The focus of this paper is the analysis of the concepts of a priori and a posteriori knowledge rather than the epistemic domain of a posteriori and a priori justification.     

有限群的一个类函数的模构造

本文研究了有限群上的一个类函数.通过计算它和不可约特征标的内积,证明了它是特征标并且通过复群代数的中心的正则表示给出了它的一个模构造.     

A REMARK ON A BMO MARTINGALE

Let M = (Mt,Ft) be a uniformly integrable continuous martingale with MO = 0. For1 5 p < cot we setIIMllBMO. = '3p II[E[IMoo ~ MTIplFT]]'/Pll.,where the supremum is taken over all stopping times T.Set BMO. = {M: IIMllBMO. < co}. It is well known that BMO. = BMO, (VI S p 5 q).F'urthermore, all 11.llBMO. norms are equivalent andIIi ~~if;llMllBMO. = SUP T P(T < co)i'where the supremum is taken over all stopping times T satisfying P(T < co) > 0. In the laterwe shall simply …     

A new characterization of A 5

Let G be a group and τ _e (G) the set of numbers of elements of G of the same order. In this paper, by τ _e (G), we give a new characterization of A ₅, where A ₅ is the alternating group of degree 5. We get the theorem following: Theorem. Let G be a group, ${G\cong A_5}$ if and only if τ _e (G) = τ _e (A ₅) = {1, 15, 20, 24}.     

NAMING A COLUMN ON A SPREADSHEET

Spreadsheets use a meaningful algebra-like notation which, research suggests, can support pupils in developing an understanding of variables. This paper discusses the activity of Year 8 pupils who were taught to name a column on a spreadsheet, and who were asked to reflect upon their activity in a stimulated recall interview. More specifically, it considers the pupils' understanding of notation, such as 'A2' and 'm', which they used when constructing spreadsheet formulae. It is suggested that experience of naming columns may help pupils to develop a clearer sense of the notation as a variable, and to make links between their spreadsheet activity and use of standard algebraic notation [1].     

A RIEMANN-HILBERT PROBLEM IN A RIEMANN SURFACE

One of the inspirations behind Peter Lax’s interest in dispersive integrable systems, as the small dispersion parameter goes to zero, comes from systems of ODEs discretizing 1-dimensional compressible gas dynamics [17]. For example, an understanding of the asymptotic behavior of the Toda lattice in different regimes has been able to shed light on some of von Neumann’s conjectures concerning the validity of the approximation of PDEs by dispersive systems of ODEs. Back in the 1990s several authors have worked on the long time asymptotics of the Toda lattice [2, 7, 8, 19]. Initially the method used was the method of Lax and Levermore [16], reducing the asymptotic problem to the solution of a minimization problem with constraints (an "equilibrium measure" problem). Later, it was found that the asymptotic method of Deift and Zhou (analysis of the associated Riemann-Hilbert factorization problem in the complex plane) could apply to previously intractable problems and also produce more detailed information. Recently, together with Gerald Teschl, we have revisited the Toda lattice; instead of solutions in a constant or steplike constant background that were considered in the 1990s we have been able to study solutions in a periodic background. Two features are worth noting here. First, the associated Riemann-Hilbert factorization problem naturally lies in a hyperelliptic Riemann surface. We thus generalize the Deift-Zhou "nonlinear stationary phase method" to surfaces of nonzero genus. Second, we illustrate the important fact that very often even when applying the powerful Riemann-Hilbert method, a Lax-Levermore problem is still underlying and understanding it is crucial in the analysis and the proofs of the Deift-Zhou method!