几种约束广义逆矩阵的有限算法

1引言与引理众所周知，关于非奇异方阵的正则逆的有限算法是由Faddeev大给在1949年之前提出的，这就是著名的Faddeev算法[1，P…334-336]。自从五十年代中期广义逆矩阵的研究复兴与发展以来，有不少学者提出了关于广义逆矩阵的有限算法。第一个给出关于广义逆矩     

Hermite-广义反Hamilton矩阵的一类反问题

给定矩阵X和B,利用矩阵的广义奇异值分解,得到了矩阵方程X~HAX=B有Hermite-广义反Hamiton解的充分必要条件及有解时解的—般表达式.用S_E表示此矩阵方程的解集合,证明了S_E中存在唯一的矩阵(?),使得(?)与给定矩阵A的差的Frobenius范数最小,并且给出了矩阵(?)的表达式;同时也证明了S_E中存在唯一的矩阵A_o,使得A_o是此矩阵方程的极小Frobenius范数Hermite-广义反Hamilton解,并且给出了矩阵A_o的表达式.     

长方矩阵常见广义逆的代数扰动

本文详细讨论了长方矩阵常见广义逆的代数扰动理论,并给出了他们代数扰动的表达式,改进了文献3,4的相应结论.     

矩阵的Γ逆

利用矩阵的广义奇异值分解,给出了复数域上矩阵的Γ逆存在的充要条件及其表达式,并讨论了Γ逆的唯-性.     

矩阵的加权Moore—Penrose逆

利用矩阵的广义奇异值分解，给出了复数域上矩阵的Moore—Penrose逆存在的充要条件及其表达式．     

广义逆矩阵的张量积

本文证明了广义逆矩阵张量积的一些性质,介绍了它在解线性方程组方面的应用,并得到了矩阵张量积的奇异值的一些性质     

矩阵的加权Moore-Penrose逆

利用矩阵的广义奇异值分解, 给出了复数域上矩阵的Moore-Penrose逆存在的充要条件及其表达式.     

广义逆矩阵的张量积

本证明了广义逆矩阵张量积的一些性质,介绍了它在解线性方程组方面的应用．并得到了矩阵张量积的奇异值的一些性质。     

对称广义中心对称矩阵的左右逆特征值问题

研究了对称广义中心对称矩阵的左右逆特征值问题,利用矩阵的奇异值分解(SVD)得到了问题的通解表达式.并由此考虑了解集合对给定矩阵的最佳逼近.     

关于两类矩阵最佳逼近问题

１．引言与引理 设Ｒｍ×ｎ表示所有ｍ×ｎ阶实矩阵的集合;ＳＲｎ×ｎ是所有ｎ阶实对称矩阵的全体;ＯＲｎ×ｎ是所有ｎ阶实正交矩阵的全体;Ｉｎ是ｎ阶单位矩阵;ＡＴ是矩阵Ａ的转置;ｒａｎｋＡ表示矩阵 Ａ的秩;‖·‖是矩阵的Ｆｒｏｂｅｎｉｕｓ范数．此外,对于 　　　　,Ａ＊Ｂ表示 Ａ与 Ｂ的 Ｈａｄａｍａｒｄ积,其定义为　　　　　　　　　　　　　,现考虑如下问题： 问题 Ⅰ给定　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　,使得 　　　　　,求 问题Ⅱ给定 ,求 ,使得 本文运用矩阵对…     

Combined perturbation bounds: Ⅱ. Polar decompositions

In this paper, we study the perturbation bounds for the polar decomposition A= QH where Q is unitary and H is Hermitian. The optimal (asymptotic) bounds obtained in previous works for the unitary factor, the Hermitian factor and singular values of A are σ2r||△Q||2F ≤ ||△A||2F,1/2||△H||2F ≤ ||△A||2F and ||△∑||2F ≤ ||△A||2F, respectively, where ∑ = diag(σ1, σ2,..., σr, 0,..., 0) is the singular value matrix of A and σr denotes the smallest nonzero singular value. Here we present some new combined (asymptotic)perturbation bounds σ2r ||△Q||2F+1/2||△H||2F≤ ||△A||2F and σ2r||△Q||2F+||△∑ ||2F ≤||△A||2F which are optimal for each factor. Some corresponding absolute perturbation bounds are also given.     

Combined perturbation bounds: Ⅱ. Polar decompositions

In this paper, we study the perturbation bounds for the polar decomposition A= QH where Q is unitary and H is Hermitian. The optimal (asymptotic) bounds obtained in previous works for the unitary factor, the Hermitian factor and singular values of A are σ2r||△Q||2F ≤ ||△A||2F,1/2||△H||2F ≤ ||△A||2F and ||△∑||2F ≤ ||△A||2F, respectively, where ∑ = diag(σ1, σ2,..., σr, 0,..., 0) is the singular value matrix of A and σr denotes the smallest nonzero singular value. Here we present some new combined (asymptotic)perturbation bounds σ2r ||△Q||2F 1/2||△H||2F≤ ||△A||2F and σ2r||△Q||2F ||△∑ ||2F ≤||△A||2F which are optimal for each factor. Some corresponding absolute perturbation bounds are also given.     

A note on the perturbation bounds of W-weighted Drazin inverse

We investigate the perturbation bound for the W-weighted Drazin inverse of a rectangular matrix and present two explicit expressions for the W-weighted Drazin inverse under the one-sided condition, which extends the results in Appl. Math. Comput. 2004;149:423–430.     

一类矩阵方程的极小Frobenius范数双对称解

利用矩阵的广义奇异值分解,给出了实矩阵方程ATXA=B存在极小Frobenius范数双对称解的充要条件及其解的表达式.     

Acute perturbation of Drazin inverse and oblique projectors

For an n×n complex matrix A with ind(A) = r; let A^D and A^π = I–AA^D be respectively the Drazin inverse and the eigenprojection corresponding to the eigenvalue 0 of A: For an n×n complex singular matrix B with ind(B) = s, it is said to be a stable perturbation of A, if I–(B^π–A^π)² is nonsingular, equivalently, if the matrix B satisfies the condition \(\mathcal{R}(B^s)\cap\mathcal{N}(A^r)=\left\{0\right\}\) and \(\mathcal{N}(B^s)\cap\mathcal{R}(A^r)=\left\{0\right\}\), introduced by Castro-González, Robles, and Vélez-Cerrada. In this paper, we call B an acute perturbation of A with respect to the Drazin inverse if the spectral radius ρ(B^π–A^π) < 1: We present a perturbation analysis and give suffcient and necessary conditions for a perturbation of a square matrix being acute with respect to the matrix Drazin inverse. Also, we generalize our perturbation analysis to oblique projectors. In our analysis, the spectral radius, instead of the usual spectral norm, is used. Our results include the previous results on the Drazin inverse and the group inverse as special cases and are consistent with the previous work on the spectral projections and the Moore-Penrose inverse.     

Sensitivity analysis for the generalized singular value decomposition

In this paper, we discuss the sensitivity of multiple nonzero finite generalized singular values and the corresponding generalized singular matrix set of a real matrix pair analytically dependent on several parameters. From our results, the partial derivatives of multiple nonzero singular values and their left and right singular vector matrices are obtained.Copyright © 2012 John Wiley & Sons, Ltd.     

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.     

一类矩阵方程的反中心对称最佳逼近解

利用矩阵的正交相似变换和广义奇异值分解,讨论了矩阵方程 AXB=C具有反中心对称解的充要条件,得到了解的具体表达式.然后应用Frobenius范数正交矩阵乘积不变性,在该方程的反中心对称解解集合中导出了与给定相同类型矩阵的最佳逼近解的表达式.     

New perturbation bounds for the subunitary polar factors

In this article, we present some new perturbation bounds for the (subunitary) unitary polar factors of the (generalized) polar decompositions. Two numerical examples are given to show the rationality and superiority of our results, respectively. In terms of the one-to-one correspondence between the weighted case and the non-weighted case, all these bounds can be applied to the weighted polar decomposition.     

On generalized inverse transversals

Let S be a regular semigroup, S° an inverse subsemigroup of S.S° is called a generalized inverse transversal of S, if V（x）∩S°≠Ф. In this paper, some properties of this kind of semigroups are discussed. In particular, a construction theorem is obtained which contains some recent results in the literature as its special cases.