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…     

矩阵方程A~TXA=D的双对称最小二乘解

１．引 言 本文用 Ｒｎ×ｍ表示全体 ｎ×ｍ实矩阵集合，用 ＳＲｎ×ｎ（ＳＲ０ｎ×ｎ）表示全体 ｎ× ｎ实对称（实对称半正定）矩阵集合，ＯＲｎ×ｎ表示全体 ｎ× ｎ实正交矩阵集合，ＢＳＲｎ×ｎ表示全体ｎ×ｎ双对称实矩阵集合．这里，一个实对称矩阵Ａ＝（ａｉｊ）ｎ×ｎ被称为双对称矩阵，如果对所有的 　　　　　　　 　　　　　　　　　　　　　　　用Ａ×Ｂ表示矩阵 Ａ与 Ｂ的Ｈａｄａｍａｒｄ乘积，Ｉｋ表示 ｋ× ｋ阶单位矩阵，Ｏ表示零矩阵，Ｓｋ＝（ｅｋ，…，ｅ２，ｅ１）∈ Ｒｋ×ｋ，其中ｅｉ表示Ｉｋ的第ｉ列． 矩阵方程…     

双对称非负定阵一类逆特征值问题的最小二乘解

１．引言 逆特征值问题在工程中有广泛的应用,其研究已有一些很好的结果［１－５］．最近,文［６］还研究了双对称矩阵逆特征值问题,即研究了如下两个问题： 问题Ａ．已知Ｘ∈Ｒｎｘｍ,Ａ＝ｄｉａｇ（λ１…,λｍ）,求Ａ∈ＢＳＲｎｘｎ使 ＡＸ＝ＸＡ,其中 Ｒｎｘｍ表示全体 ｎ ｘ ｍ实矩阵集合, ＢＳＲｎｘｎ表示全体 ｎ ｘ ｎ双对称阵集合． 问题Ｂ．已知Ａ＊ＥＲｎｘｎ,求Ａ∈ＳＥ使 ｜｜Ａ＊－Ａ｜｜＝ ｉｎｆ ｜｜Ａ＊－Ａ｜｜ ＡＦＳＥ其中 ＳＥ是问题 Ａ的解集合,｜｜． ｜｜表示 Ｆｒｏｂｅｎｉｕｓ范数． 在实际问题中, …     

一类双对称矩阵反问题的最小二乘解

１．问题的提出近年来,对于矩阵反问题ＡＸ＝Ｂ的研究已取得了一系列的结果［１］,获得了解存在的条件,但由于实际问题中Ｘ,Ｂ由实验给出,很难保证满足解存在的条件,因此研究问题的最小二乘解是有实际意义的．本文就结构设计中用到的一类双对称矩阵的最小二乘问题进行探讨．令Ｒ~(ｎ×ｍ)表示所有ｎ×ｍ阶实矩阵集合,Ｒ~n=Ｒ~(n×1)　表示其中秩为ｒ的子集;ＯＲ~（ｎ×ｎ）　表示所有ｎ阶正交阵之集;Ａ~（ ）表示矩阵Ａ的Ｍｏｏｒｅ－Ｐｅｎｒｏｓｅ广义逆;Ｉ_ｋ表示ｋ阶单位阵;||·||表示Fｒｏｂｅｎｉｕｓ范数;表示ＳＲ~(ｎ…     

一类亚半正定矩阵的左右逆特征值问题

1．引言在工程技术中常常遇到这样一类逆特征值问题：要求在一个矩阵集合S中，找具有给定的部分右特征对（特征值及相应的特征向量）和给定的部分左特征对（特征值及相应的特征向量）的矩阵．文［2］，［3］讨论了S为。x。实矩阵集合的情形．文［4］－［7］对S为nxn实对称矩阵．对称正定矩阵，对称半正定矩阵集合的情形进行了讨论．文【川讨论了S为亚正定阵集合的情形．并提到了对于亚半正定矩阵的情形目下无人涉及，有待进一步研究．本文将对S为nxn亚半正定矩阵集合的情形进行讨论．给出了亚半正定矩阵的左右逆特征值问题有解的充要条件…     

线性流形上的矩阵最佳逼近

令Ｓ＝｛Ａ∈Ｒｎ×ｍ｜ｆ１（Ａ）＝‖ＡＸ１－Ｚ１‖２＋‖ＹＴ１Ａ－ＷＴ１‖２＝ｍｉｎ｝，其中Ｘ１∈Ｒｍ×ｋ１，Ｚ１∈Ｒｎ×ｋ１，Ｙ１∈Ｒｎ×１１和Ｗ１∈Ｒｍ×１１均为给定的矩阵，‖·‖是Ｆｒｏｂｅｎｉｕｓ范数。本文考虑如下问题：问题Ⅰ给定Ｘ２∈Ｒｍ×ｋ２，Ｚ２∈Ｒｎ×ｋ２，Ｙ２∈Ｒｎ×ｌ２，Ｗ２∈Ｒｍ×ｌ２，求Ａ∈Ｓ，使得ｆ２（Ａ）＝‖ＡＸ２－Ｚ２‖２＋‖ＹＴ２Ａ－ＷＴ２‖２＝ｍｉｎ．问题Ⅱ给定Ａ∈Ｒｎ×ｍ，求Ａ∈ＳＡ，使得‖Ａ－Ａ‖＝ｉｎｆＡ∈ＳＡ‖Ａ－Ａ‖，其中ＳＡ是问题Ｉ的解集合。本文给出问题Ｉ解集合ＳＡ的通式和问题Ⅱ的解Ａ的表达式，提出了求解问题Ⅰ与Ⅱ的数值方法。许多文献的结果都是本文结果的特例。     

四元数矩阵方程AX=B的实部半正定解(英文)

一个ｎ×ｎ实四元数矩阵称为实部半正定（或正定）矩阵，如果对于任意的非零ｎ维四元数列向量ｘ，有Ｒｅ［ｘＡｘ］≥０（或＞０）．本文给出了四元数矩阵方程ＡＸ＝Ｂ有实部半正定（或正定）矩阵解的充要条件及其通解的表达式，并给出了四元数分块阵为实部半正定（或正定）矩阵的一个判别法则     

关于实方阵的几个问题

本文讨论如下内容：１．把有关对称正定（半正定）的一些性质推广到广义正定（半正定）。２．给定ｘ∈Ｒｍ×ｍ，∧为对角阵，求ＡＸ＝ｘ∧在对称半正定矩阵类中解存在的充要条件及一般形式，并讨论了对任意给定的对称正定（半正定）矩阵Ａ，在上述解的集合中求得Ａ，使得     

Lavoie不等式的改进

设为ｍ×ｎ复数阵，Ａ．为ｍｉ×ｎ阵，记Ｂ＝其中表示Ａｉ的Ｍｏｏｒｅ－Ｐｅｎｒｏｓｅ广义逆。Ｌａｖｏｉｅ［１］证明了不等式：０≤ｄｅｔＡＢ≤１．本文的目的是对这个不等式作进一步改进。     

恰含d个非零对角元的本原矩阵的广义最大密度指数集

设Ａ是一个具有周期ｐ的ｎ×ｎ不可约布尔矩阵,文［１］定义了矩阵的广义最大密度指数ｈＡ（ｋ）令ＤＩＳｎ,ｄ（ｋ）＝｛ｈＡ（ｋ）｜ Ａ ＰＭｎ（ｄ）｝,其中ＰＭｎ（ｄ）是所有恰含ｄ个非零对角元的ｎ×ｎ本原矩阵的集合．本文证明了另外,我们定义矩阵Ａ的范数,用Ａ表示,为Ａ中１的个数,并且刻划了具有最小范数的极矩阵．     

流形上矩阵方程AX=B的正定及半正定阵的反问题

文[1-5]中研究了对称、对称半正定及流形上的对称半正定的反问题,并说明了其应用背景.本文研究线性流形上的正定及半正定阵的反问题,说明了文[1-3]中的一些结果为本文的特例.     

Characterizations of minimal semipositivity

A real m×n matrix A is said to be semipositive if there is a nonnegative vector λ such that Ax exists and is componentwise positive. A is said to be minimally semipositive if it is semipositive and no proper m×p submatrix of A is semipositive. Minimal semipositivity is characterized in this paper and is related to rectangular monotonicity and weak r-monotonicity. P-matrices and nonnegative matrices will also be considered.     

New results about semi-positive matrices

Our purpose is to present a number of new facts about the structure of semipositive matrices, involving patterns, spectra and Jordon form, sums and products, and matrix equivalence, etc. Techniques used to obtain the results may be of independent interest. Examples include: any matrix with at least two columns is a sum, and any matrix with at least two rows, a product, of semipositive matrices. Any spectrum of a real matrix with at least 2 elements is the spectrum of a square semipositive matrix, and any real matrix, except for a negative scalar matrix, is similar to a semipositive matrix. M-matrices are generalized to the non-square case and sign patterns that require semipositivity are characterized.     

关于矩阵张量积的一类问题

本文给出有限个矩阵张量积分别是正规矩阵、厄米特矩阵、正定矩阵的条件．推广了Y．E．Kuo的相关结果．另外也给出了两个亚半正定矩阵的张量积还是亚半正定矩阵的充要条件．     

Semipositive matrices and their semipositive cones

The semipositive cone of \(A\in \mathbb {R}^{m\times n}, K_A = \{x\ge 0\,:\, Ax\ge 0\}\), is considered mainly under the assumption that for some \(x\in K_A, Ax>0\), namely, that A is a semipositive matrix. The duality of \(K_A\) is studied and it is shown that \(K_A\) is a proper polyhedral cone. The relation among semipositivity cones of two matrices is examined via generalized inverse positivity. Perturbations and intervals of semipositive matrices are discussed. Connections with certain matrix classes pertinent to linear complementarity theory are also studied.

     

Symmetric period solutions with prescribed minimal period for even autonomous semipositive Hamiltonian systems

We study some monotonicity and iteration inequality of the Maslov-type index i-1of linear Hamiltonian systems.As an application we prove the existence of symmetric periodic solutions with prescribed minimal period for first order nonlinear autonomous Hamiltonian systems which are semipositive,even,and superquadratic at zero and infinity.This result gives a positive answer to Rabinowitz’s minimal period conjecture in this case without strictly convex assumption.We also give a different proof of the existence of symmetric periodic solutions with prescribed minimal period for classical Hamiltonian systems which are semipositive,even,and superquadratic at zero and infinity which was proved by Fei,Kim and Wang in 2001.     

关于“四元数自共轭矩阵与行列式的几个定理”的注记

本刊1985年第4期发表郝稚传的“四元数自共轭矩与行列式的几个定理”(称为〔1〕)的文章的主要工作分为两部分。〔1〕在英文摘要中写到:“(i)This essay has improvedthe conclusion of theorem 8 and theorem 9 in 〔2〕”即〔1〕之第一部分工作在于改进了〔2〕的定理8、9的结果,其根据就是〔1〕的定理1、2.〔1〕称定理1—若     

Algebraic eigenspaces of nonnegative matrices

The Perron-Frobenius theory for square, irreducible, nonnegative matrices is generalized by studying the structure of the algebraic eigenspace of an arbitrary square nonnegative matrix corresponding to its spectral radius. We give a constructive proof that this subspace is spanned by a set of semipositive vectors and give a combinatorial characterization of both the index of the spectral radius and dimension of the algebraic eigenspace corresponding to the spectral radius. This involves a detailed study of the standard block triangular representation of nonnegative matrices by giving special attention to those blocks on the diagonal having the same spectral radius as the original matrix. We also show that the algebraic eigenspace corresponding to the spectral radius contains a semipositive vector having the largest set of positive coordinates among all vectors in this subspace.     

Equilibrants, semipositive matrices, calculation and scaling

For square, semipositive matrices A (Ax>0 for some x>0), two (nonnegative) equilibrants e(A) and E(A) are defined. Our primary goal is to develop theory from which each may be calculated. To this end, the collection of semipositive matrices is partitioned into three subclasses for each equilibrant, and a connection to those matrices that are scalable to doubly stochastic matrices is made. In the process a certain matrix/vector equation that is related to scalability of a matrix to one with line sums 1 is derived and discussed.     

Stochastic observability and applications

In this paper the problem of stochastic observability of alinear system affected by multiplicative white noise and Markovianjumping is investigated. The definition of stochastic observabilityadopted here extends to this framework the definition of thewell known uniform observability of a deterministic time-varyinglinear system. By several examples we show that the conceptof stochastic observability introduced in this paper is lessrestrictive than those introduced in other existing works andit does not always imply stochastic detectability as would beexpected. Finally we prove that this kind of stochastic observabilityallows us to derive a Barbasin–Krasovskii type resultfor exponential stability in mean square. This provides a sufficientcondition which guarantees that any semipositive solution ofcorresponding Riccati differential equation is a stabilizingsolution.