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半正定分块矩阵和一个线性矩阵方程及其反问题 总被引:6,自引:0,他引:6
一个实的(未必对称)n×n矩阵A称为半正定的,如果对任意非零的n维行向量x,均有xMxt≥0.本文给出了一个分块n×n矩阵为半正定的充要条件.另外,我们讨论了线性矩阵方程AX=B对解附加种种条件下的解.我们应用矩阵在相抵下的标准形给出了这一方程的相容性的充要条件.还给出这个方程的反问题在对解附加各种条件下的解. 相似文献
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矩阵方程A^TXB=C的正定和半正定解 总被引:4,自引:1,他引:4
何楚宁 《高校应用数学学报(A辑)》1997,(4):475-480
给出了矩阵方程A^TXB=C在正定和半正定矩阵类中有解的充要条件及解的一般表达式。 相似文献
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非对称半正定矩阵的一些性质阳本傅(成都师范高等专科学校数学系611930)设A是n阶实矩阵(不一定对称),如果对任意实n元向量X,均有X′AX0(>0),就称A为半正定矩阵(正定矩阵).本文给出半正定矩阵的一种合同标准形,由此比较简捷地得出了半正定... 相似文献
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一类亚半正定矩阵的左右逆特征值问题 总被引:8,自引:0,他引:8
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… 相似文献
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半正定矩阵及矩阵方程AX=B的反问题 总被引:8,自引:0,他引:8
文从研究一类控制系统的实际背景提出对已知实向量x,b求满足Ax=b的对称正定阵A的一类反问题。文[2]与[3]研究了上述反问题在对称正定类、正定类中有解的充要条件及解的一般形式。本文讨论复矩阵方程 AX=B(1)(X,B为m×n阵,A为m×m阵)在半正定、正定、H半正定、H正定类中反问题有解的充要条件及其解集的一般形式。如无特别申明,本文总考虑复矩阵和复向量,其共轭转置用“*”表 相似文献
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This paper focuses on L-structured quaternion matrices. L-structured real matrices, conditions for the existence of solutions and the general solution of linear matrix equations were studied in the paper [Magnus JR. L-structured matrices and linear matrix equations, Linear Multilinear Algebra 1983;14:67–88]. In this paper, we present a theoretical study extending L-structured real matrices to L-structured quaternion matrices, and introduce some L-structured quaternion matrices. Based on them, we then discuss their applications in quaternion matrix equations. 相似文献
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In this article, we present a weaker version of the class of generalized positive subdefinite matrices introduced by Crouzeix and Komlósi [J.P. Crouzeix and S. Komlósi, The Linear Complementarity Problem and the Class of Generalized Positive Subdefinite Matrices, Applied Optimization, Vol. 59, Kluwer, Dordrecht, 2001, pp. 45–63], which is new in the literature, and obtain some properties of weak generalized positive subdefinite (WGPSBD) matrices. We show that this weaker class of matrices is also captured by row-sufficient matrices introduced by Cottle et al. [R.W. Cottle, J.S. Pang, and V. Venkateswaran, Sufficient matrices and the linear complementarity problem, Linear Algebra Appl. 114/115 (1989), pp. 231–249] and show that for WGPSBD matrices under appropriate assumptions, the solution set of a linear complementarity problem is the same as the set of Karush–Kuhn–Tucker-stationary points of the corresponding quadratic programming problem. This further extends the results obtained in an earlier paper by Neogy and Das [S.K. Neogy and A.K. Das, Some properties of generalized positive subdenite matrices, SIAM J. Matrix Anal. Appl. 27 (2006), pp. 988–995]. 相似文献
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吴炎 《纯粹数学与应用数学》2012,(2):155-166
设R是2为单位的局部环.研究了R上三个两两可换的n阶非零幂等矩阵的线性组合广义逆之间的包含关系,确定了R上一类特殊矩阵广义逆的列表算法.利用这种列表算法和相关的矩阵理论,得到了这些矩阵线性组合广义逆之间的包含关系的充要条件,推广了矩阵自反广义逆的逆反律的相关结果. 相似文献
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A (0, 1) matrix is linear if it does not contain a 2×2 submatrix of all ones. In this paper we give polynomial algorithms to test whether a linear matrix is balanced or perfect. The algorithms are based on decomposition results previously obtained by the authors.Partial support under NSF Grants DMS 8606188, DDM8800281 and DDM9001705. 相似文献
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本文选择列随机平均矩阵T_n作为基本代换矩阵,建立了基于T_n的逐次差分代换方法.获得了R_+~n上正半定型,不定型判定的充要条件.并进一步证明了:正定型的差分代换集序列正向终止.根据这些结果编写的Maple程序TSDS3,能够自动证明代数型不等式,对不成立的不等式总能输出反例.该程序虽可能不停机,但大量的应用实例证实了该方法的实用性. 相似文献
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给定广义自反矩阵R,S,即R=R=R-1,S=S=S-1,若复矩阵X满足条件RXS=X(或RXS=X),则称其为(R,S)-对称矩阵(或(R,S)-斜对称矩阵).分别讨论了线性流形上(R,S)-对称矩阵和(R,S)-斜对称矩阵约束下矩阵方程MZN=E的最小二乘问题,得到了通解表达式. 相似文献
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We prove two basic conjectures on the distribution of the smallest singular value of random n×n matrices with independent entries. Under minimal moment assumptions, we show that the smallest singular value is of order n−1/2, which is optimal for Gaussian matrices. Moreover, we give a optimal estimate on the tail probability. This comes as a consequence of a new and essentially sharp estimate in the Littlewood-Offord problem: for i.i.d. random variables Xk and real numbers ak, determine the probability p that the sum k∑akXk lies near some number v. For arbitrary coefficients ak of the same order of magnitude, we show that they essentially lie in an arithmetic progression of length 1/p. 相似文献
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Alexander Barvinok 《Advances in Mathematics》2010,224(1):316-757
We consider the set Σ(R,C) of all m×n matrices having 0-1 entries and prescribed row sums R=(r1,…,rm) and column sums C=(c1,…,cn). We prove an asymptotic estimate for the cardinality |Σ(R,C)| via the solution to a convex optimization problem. We show that if Σ(R,C) is sufficiently large, then a random matrix D∈Σ(R,C) sampled from the uniform probability measure in Σ(R,C) with high probability is close to a particular matrix Z=Z(R,C) that maximizes the sum of entropies of entries among all matrices with row sums R, column sums C and entries between 0 and 1. Similar results are obtained for 0-1 matrices with prescribed row and column sums and assigned zeros in some positions. 相似文献