一类Stiefel流形上极小化问题的低秩解

<正>1引言记冗R~(m×n)为m×n阶实数矩阵集合;A~T表示矩阵A的转置;I_p表示p×p阶单位矩阵.对任意矩阵A=(a_(ij))∈R~(m×n),[A]_(ij)表示A的第ij个元素,即[A]_(ij)=a_(ij);‖A‖_F表示矩阵A的Frobenius范数,且有关系‖A‖_F~2=tr(A~TA),(1.1)其中tr(·)表示矩阵的迹,且有性质tr(A+B)=tr(A)+tr(B),tr(AB)=tr(BA),tr(B~T)=tr(B).(1.2)本文研究如下Stiefel流形上的极小化问题:     

关于矩阵范数的几个不等式

正1引言文中,用M_n表示n×n复矩阵全体,用‖·‖表示任意的酉不变范数,分别用|λ_n(A)|≤…≤|λ_1(A)|,s_n(A)≤…≤s1(A)来表示矩阵A的特征值和奇异值,用|A|=(A~*A)~(1/2)表示A的绝对值算子.     

关于积和式差的一点注记

It is shown that for any two n×n complex valued matrices A, B the inequality |perA-perB|≤n‖A-B‖Fmax(‖A‖F,‖B‖F)n-1 or |perA-perB|≤‖A‖Fn ‖B‖Fn holds for where AH denotes the conjuagte transpose of the matrix A=(aij)n×n.     

线性流形上矩阵方程BTXB=F的双对称最小二乘解

1引言令R~(n×m)、OR~(n×n)、SR~(n×n)(SR_0~(n×n))分别表示所有n×m阶实矩阵、n阶实正交阵、n阶实对称矩阵(实对称半正定阵)的全体,A~ 表示A的Moore-Penrose广义逆,I_k表示k阶单位矩阵,S_k表示k阶反序单位矩阵。R(A)表示A的列空间,N(A)表示A的零空间,rank(A)表示矩阵A的秩。对A=(a_(ij)),B=(b_(ij))∈R~(n×m),A*B表示A与     

正规矩阵的任意扰动

设A为n×n矩阵,其特征值为λ1,λ2,…,λn;矩阵B=A+X之特征值为μ1,μ2,…,μn.若A,B均为正规矩阵,由Wielandt-Hoffman定理[1],存在1,2,…,n的一个排列k1,k2,…,kn,使得nj=1|λj-μkj|2≤‖X‖2F,(1)其中‖·‖F表示Frobenius范数.又,在同样条件下,存在1,2,…,n的一个排列l1,l2,…,ln,使得对1≤j≤n均有|λj-μlj|≤2.91‖X‖2,(2)其中‖·‖2表示谱范数,这是R.Bhatia等人的结果[2].本文旨在讨论A为正规矩阵,B为任意矩阵时特征值的扰动估计,得到了几个扰动定理,分别推广了上述两个结果.本文用CH表示矩阵C的共轭转置,trC表示C的迹;…     

几类矩阵范数之间的关系

1 引言 [1]中,讨论了C~(m×n)上的矩阵A的L_p范数/A/_p=(∑/a_(ij)/~p)和l_p算子范数‖A‖_p= max/AX/_p1』之间的关系,得到了下面的不等式: ‖A‖_p≤μ_p(n)/A/_p, ‖A‖_p≤μ_q(m)/A/_p, (1.1) 这里     

矩阵方程AXB+CYD=E对称最小范数最小二乘解的极小残差法

<正>1引言本文用R~(n×m)表示全体n×m实矩阵集合,用SR~(n×n)表示全体n×n实对称矩阵集合,OR~(n×n)表示全体n×n实正交矩阵集合.用I_n表示n阶单位矩阵,用A*B表示矩阵A与B的Hadamard乘积.对任意矩阵A,B∈R~(n×m),定义内积〈A,B〉=tr(B~T A),其中     

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.     

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.     

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.     

A parameterized hessian quadratic programming problem

We present a general active set algorithm for the solution of a convex quadratic programming problem having a parametrized Hessian matrix. The parametric Hessian matrix is a positive semidefinite Hessian matrix plus a real parameter multiplying a symmetric matrix of rank one or two. The algorithm solves the problem for all parameter values in the open interval upon which the parametric Hessian is positive semidefinite. The algorithm is general in that any of several existing quadratic programming algorithms can be extended in a straightforward manner for the solution of the parametric Hessian problem.This research was supported by the Natural Sciences and Engineering Research Council under Grant No. A8189 and under a Postgraduate Scholarship, by an Ontario Graduate Scholarship, and by the University of Windsor Research Board under Grant No. 9432.     

A parameterized hessian quadratic programming problem

We present a general active set algorithm for the solution of a convex quadratic programming problem having a parametrized Hessian matrix. The parametric Hessian matrix is a positive semidefinite Hessian matrix plus a real parameter multiplying a symmetric matrix of rank one or two. The algorithm solves the problem for all parameter values in the open interval upon which the parametric Hessian is positive semidefinite. The algorithm is general in that any of several existing quadratic programming algorithms can be extended in a straightforward manner for the solution of the parametric Hessian problem. This research was supported by the Natural Sciences and Engineering Research Council under Grant No. A8189 and under a Postgraduate Scholarship, by an Ontario Graduate Scholarship, and by the University of Windsor Research Board under Grant No. 9432.     

一类矩阵方程的埃尔米特自反最小二乘解

利用埃尔米特自反矩阵的表示定理和矩阵的拉直方法,研究了矩阵方程$AX+BY=C$的埃尔米特自反最小二乘问题,进一步,给出了方程在埃尔米特自反矩阵集合中可解的充分必要条件,得到解的一般表达式,最后,对任意给定的一对复矩阵,得到了其相关最佳逼近问题解的表达式.     

ON HASEMAN BOUNDARY VALUE PROBLEM FOR A CLASS OF META-ANALYTIC FUNCTIONS

In this article, Haseman boundary value problem for a class of meta-analytic functions is studied. The expression of solution and the condition of solvability for Haseman boundary value problem are obtained by changing the problem discussed into the equivalent Haseman boundary value problem of bi-analytic function. And the expression of solution and the condition of solvability depend on the canonical matrix.     

由三个特征对构造正定Jacobi矩阵

本文研究了由三个特征对构造正定Jacobi矩阵的问题,给出了这个问题有唯一解的充要条件及解的表达式,并给出了问题的数值算法.     

A structure‐preserving doubling algorithm for Lur'e equations

We introduce a numerical method for the numerical solution of the Lur'e equations, a system of matrix equations that arises, for instance, in linear‐quadratic infinite time horizon optimal control. We focus on small‐scale, dense problems. Via a Cayley transformation, the problem is transformed to the discrete‐time case, and the structural infinite eigenvalues of the associated matrix pencil are deflated. The deflated problem is associated with a symplectic pencil with several Jordan blocks of eigenvalue 1 and even size, which arise from the nontrivial Kronecker chains at infinity of the original problem. For the solution of this modified problem, we use the structure‐preserving doubling algorithm. Implementation issues such as the choice of the parameter γ in the Cayley transform are discussed. The most interesting feature of this method, with respect to the competing approaches, is the absence of arbitrary rank decisions, which may be ill‐posed and numerically troublesome. The numerical examples presented confirm the effectiveness of this method. Copyright © 2015 John Wiley & Sons, Ltd.     

有资格限制的指派问题的求解方法

在实际的指派工作中，常会遇到某个人有没有资格去承担某项工作的问题，因此，本建立了有资格限制的指派问题的数学模型。在此数学模型中，将效益矩阵转化为判定矩阵，由此给出了判定此种指派问题是否有解的方法；在有解的情况下，进一步将效益矩阵转化为求解矩阵，从而将有资格限制的指派问题化为传统的指派问题来求解。最后给出了一个数值例子来说明这样的处理方法是有效的。     

矩阵不等式约束下矩阵方程AX=B的双对称解

本文讨论矩阵不等式CXD≥E 约束下矩阵方程AX=B的双对称解,即给定矩阵A,B,C,D和 E, 求双对称矩阵X, 使得AX=B 和 CXD≥E, 其中CXD≥E表示矩阵CXD-E非负.本文将问题转化为矩阵不等式最小非负偏差问题,利用极分解理论给出了求其解的迭代方法,并结合相关矩阵理论说明算法的收敛性.最后给出数值算例验证算法的有效性.     

通解矩阵(英文)

用消元法程序在计算机上求含ｍ个方程ｎ个未知量的线性代数方程组的通解的问题现在仍然没有得到解决．本文首先给出一个称为“通解矩阵”的新定义，然后证明两个有关定理．以此为基础，使解决上述问题成为一件容易的事．     

由主子阵和特殊次序缺损特征对构造Jacobi矩阵

In this paper,an inverse eigenvalue problem of constructing a Jacobian matrix from its prescribed specially ordered defective eigenpairs and a principal subma-trix is considered.The necessary and sufficient conditions for the existence and uniqueness of the solution are derived.Two numerical algorithms and two numer-ical examples are given.