广义极分解

本文使用下列符号:C~(m×n)表示m×n复矩阵的集合,C_r~(m×n)表示秩为r的m×n复矩阵的集合,A~H和A~+分别表示矩阵A的共轭转置和Moore-Penrose广义逆,|| ||_2表示向量的Euclid范数和矩阵的谱范数,|| ||_F表示Frobenius范数,R(A)表示A的列     

矩阵反问题解的稳定性

首先说明一些记号.C~(m×n):所有m×n复元素矩阵的全体,C_r~(m×n):C~(m×n)中所有秩为r的矩阵的全体.A~H:矩阵A的转置共轭.I~((n)):n行列单位矩阵.A>0表示A是正定Hermite矩阵,λ_(max)(A)与λ_(min)(A)分别表示Hermite矩阵A的最大与最小特征值,σ_(max)(A)与σ_(min)(A)分别表示矩阵A的最大与最小奇异值.A~+:A的Moors-Penrose广义逆.|| ||_2:矩阵的谱范数,|| ||_F:矩阵的Frobenius范数.     

矩阵双侧旋转与逼近

首先,引入一些符号.用C~(m×n)表示m×n复矩阵的集合,U~(i×m)={A∈C~(l×m)│A~(II)A=I_m(l≥m)}.I_m表示m阶单位矩阵,A~H表示矩阵A的共轭转置矩阵,tr(A)表示矩阵A的迹,Re[tr     

离散李雅普诺夫矩阵方程X-AXB=C的误差界

1 引 言 以C~(m×n)表所有m×n复元素矩阵的全体,对于给定的矩阵A∈C~(m×m),B∈C~(n×n)和C∈C~(m×n),矩阵方程 X-AXB=C (1.1)称为离散李雅普诺夫矩阵方程,它与控制理论有密切的关系。关于这类方程的解法,     

广义逆与最优化

目前,广义逆在最优化中得到越来越多的应用,广义逆成了研究最优化的一个重要和有效的工具.最优化中的许多问题可以利用广义逆给出清晰、本质的表示.最优化中的病态问题(包括奇异性问题),可以通过考虑广义逆矩阵得到解决.本文按照作者的观点综述了广义逆矩阵在最优化各个领域中的应用.在本文中,我们用 R~m(C~m)表示 m 维向量空间,R~(m×n)(C~(m×n)表示 m×n 矩阵的     

任意体上矩阵的ρMoore-Penrose逆的某些显式

设K是一个任意的体,表示K上所有矩阵的集合,K~(m×n)表示K上m×n矩阵的集合,K_r~(m×n)={A∈K~(m×n)|RankA=r}.推广[1]中的概念,我们引入定义1.设的一个变换,如果满足 (i)(AB)~ρ=B~ρA~ρ,A∈K~(m×n),B∈K~(?); (ii)(A~ρ)~ρ=A,A∈, 那么ρ叫做的一个对合函数. 定义2.设ρ是的一个对合函数,A∈K~(m×n),如果存在X∈K~(n×m),满足下面关于ρ的Penrose方程:     

分块矩阵取得极值秩的条件

证明了如何选取矩阵X,Y和Z使得下面的分块矩阵(AXYZ)取得它的极大秩和极小秩,这里A∈C~(m×n)是一个已知矩阵,X∈C~(m×k),Y∈C~(p×n)和Z∈C~(p×k)是三个任意矩阵.     

线性流形上Hermite-广义反Hamilton矩阵反问题的最小二乘解

1.引言 令Rn×m表示所有n×m实矩阵集合,Cn×m表示所有n×m复矩阵集合,Cn=Cn×1,HCn×n表示所有n阶Hermite矩阵集合,UCn×n表示所有n阶酉矩阵集合,AHCn×n表示所有n阶反Hermite矩阵集合,R(A)表示A的列空间,N(A)表示A的零空间,A+表示A的Moore—Penrose广义逆,A*B表示A与B的Hadamard积,rank(A)表示矩阵A的秩.tr(A)表示矩阵A的迹.矩阵A,B的内积定义为(A,B)=tr(BHA),A,B∈Cn×m,由此内积诱导的范数为||A||=√(A,A)=[tr(AHA)]1/2,则此范数为Frobenius范数,并且Cn×m构成一个完备的内积空间,In表示n阶单位阵,i=√-1,记OASRn×n表示n×n阶正交反对称矩阵的全体,即     

实对称带状矩阵特征值反问题

用R~(n×m)表示所有n×m实矩阵的集合;OR~(n×n)表示所有n×n正交矩阵的集合;S_(n,r)表示所有带宽为2r+1的n阶实对称矩阵的集合;||·||_F表示矩阵的Frobenius范数,||·||表示向量的Euclid范数.任取A∈R~(n×m),满足AA~-A=A 的A~-∈R~(m×n)叫做A的内逆,满足AA_l~-A=A和(AA_l~-)~T=AA_l~-的A_l~-∈R~(m×n)叫做A的最小二乘广义逆,     

广义逆矩阵的连续性问题——数值相关性理论的应用

如所周知,m×n阶矩阵A的Moore—Penrose广义逆在A不满秩时是不连续的。本文证明,这种不连续性不是本质的,经保秩变形后就自动消失了。     

Möbius Inversion of Random Acyclic Directed Graphs

Suppose a random acyclic digraph has adjacency matrix A with independent columns or independent rows. Then the mean Möbius inverse of the zeta matrix I + A is the Möbius inverse of the mean zeta matrix, i.e., E[(I + A )^?1]=[I + E( A )]^?1.     

Hermite矩阵方程

本文讨论矩阵方程X^*AX=A的求解,其中A为Hermite矩阵,X^*为X的转置矩阵.文中给出解的表示式.     

Computing a matrix symmetrizer exactly using modified multiple modulus residue arithmetic

A symmetric solution X satisfying the matrix equation XA = A^tX is called a symmetrizer of the matrix A. A general algorithm to compute a matrix symmetrizer is obtained. A new multiple-modulus residue arithmetic called floating-point modular arithmetic is described and implemented on the algorithm to compute an error-free matrix symmetrizer.     

关于圈的完全正矩阵实现

本文给出了一个关联图为圈的非负、半正定矩阵A为完全正的一个充要条件.我们还证明了这样的矩阵A（当A为完全正时）的分解指数即为A的阶数.     

Inclusion sets for the singular values of a square matrix

The paper presents a general approach to deriving inclusion sets for the singular values of a matrix A = (a_ij) ∈ ℂ ^n×n. The key to the approach is the following result: If σ is a singular value of A, then a certain matrix C(σ, A) of order 2n, whose diagonal entries are σ² − | a_ii|², i = 1, …, n, is singular. Based on this result, we use known diagonal-dominance type nonsingularity conditions to obtain inclusion sets for the singular values of A. Scaled versions of the inclusion sets, allowing one, in particular, to obtain Ky Fan type results for the singular values, are derived by passing to the conjugated matrix D⁻¹C(σ, A)D, where D is a positive-definite diagonal matrix. Bibliography: 16 titles. Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 359, 2008, pp. 52–77.     

The solvability conditions for inverse eigenproblem of symmetric and anti‐persymmetric matrices and its approximation

The problem of generating a matrix A with specified eigen‐pair, where A is a symmetric and anti‐persymmetric matrix, is presented. An existence theorem is given and proved. A general expression of such a matrix is provided. We denote the set of such matrices by ??????_Eⁿ. The optimal approximation problem associated with ??????_Eⁿ is discussed, that is: to find the nearest matrix to a given matrix A* by A∈??????_Eⁿ. The existence and uniqueness of the optimal approximation problem is proved and the expression is provided for this nearest matrix. Copyright © 2002 John Wiley & Sons, Ltd.     

On the matrix inequality (I + X)−1 ≤ I

In this note we consider the question under which conditions all entries of the matrix I???(I?+?X)^?1 are nonnegative in case matrix X is a real positive definite matrix. Sufficient conditions are presented as well as some necessary conditions. One sufficient condition is that matrix X ^?1 is an inverse M-matrix. A class of matrices for which the inequality holds is presented.     

Fast and Stable Reduction of Diagonal Plus Semi-Separable Matrices to Tridiagonal and Bidiagonal Form

A two-way chasing algorithm to reduce a diagonal plus a symmetric semi-separable matrix to a symmetric tridiagonal one and an algorithm to reduce a diagonal plus an unsymmetric semi-separable matrix to a bidiagonal one are considered. Both algorithms are fast and stable, requiring a computational cost of N ², where N is the order of the considered matrix.     

Minimal paramount matrices

A real symmetric matrix of order n, n ? 2, is said to be paramount if each proper principal minor is not less than the absolute value of any other minor built from the same rows. A paramount matrix is minimal ¹ if reducing any of the diagonal entries removes the matrix from the paramount class. Minimal paramount matrices arise in the n-port realization problem of circuit theory. A condition is found that is equivalent to the minimality of a paramount matrix. Conditions are also found that guarantee that the inverse of an invertible minimal paramount matrix is itself minimal.     

The structured distance to normality of Toeplitz matrices with application to preconditioning

A formula for the distance of a Toeplitz matrix to the subspace of {e^i?}‐circulant matrices is presented, and applications of {e^i?}‐circulant matrices to preconditioning of linear systems of equations with a Toeplitz matrix are discussed. Copyright © 2010 John Wiley & Sons, Ltd.