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浙江大学学报（理学版）
浙江大学学报（理学版）  2017, Vol. 44 Issue (5): 511-515    DOI: 10.3785/j.issn.1008-9497.2017.05.002
数学与计算机科学     
一类矩阵特征值的不等式及其在Fischer不等式证明中的应用
张华民1, 殷红彩2
1. 蚌埠学院 数理系, 安徽 蚌埠 233030;
2. 安徽财经大学 管理科学和工程学院, 安徽 蚌埠 233000
An eigenvalue inequality of a class of matrices and its applications in proving the Fischer inequality
ZHANG Huamin1, YIN Hongcai2
1. Department of Mathematics & Physics, Bengbu University, Bengbu 233030, Anhui Province, China;
2. School of Management Science and Engineering, Anhui University of Finance & Economics, Bengbu 233000, Anhui Province, China
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摘要： Hadamard和Fischer不等式在矩阵研究中起重要作用.已有大量文献研究此两不等式的新证明、推广、细化及应用.本文研究了和实对称正定矩阵相关的一类矩阵的特征值，并建立了关于这类矩阵特征值乘积范围的一个不等式，利用此不等式证明了行列式的Fischer和Hadamard不等式.
关键词： 正定矩阵;  特征值;  特征向量;  行列式不等式    
Abstract: The Hadamard inequality and Fischer inequality play an important role in the matrix study. Many articles have addressed these inequalities providing new proofs, noteworthy extensions, generalizations, refinements, counterparts and applications. This paper discusses the eigenvalues of a class of matrices related to the real symmetric positive definite matrix and establishes an inequality of the eigenvalues. By using this inequality, the Fischer determinant inequality and Hadamard determinant inequality are proved.
Key words: positive definite matrix    eigenvalue    eigenvector    determinant inequality
收稿日期: 2016-02-04 出版日期: 2017-05-01
CLC:  O151.2  
基金资助: Supported by Natural Science Foundation of Anhui Provincial Education Department (KJ2016A458) and Excellent Personnel Domestic Visiting Project (gxfxZD2016274).
作者简介: 张华民(1972-),ORCID:http://orcid.org/0000-0002-7416-7415,male,doctor,associate professor,the field of interest are matrix theory and its applications,E-mail:zhangeasymail@126.com.
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引用本文:

张华民, 殷红彩. 一类矩阵特征值的不等式及其在Fischer不等式证明中的应用[J]. 浙江大学学报（理学版）, 2017, 44(5): 511-515.

ZHANG Huamin, YIN Hongcai. An eigenvalue inequality of a class of matrices and its applications in proving the Fischer inequality. Journal of Zhejiang University (Science Edition), 2017, 44(5): 511-515.

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https://www.zjujournals.com/sci/CN/10.3785/j.issn.1008-9497.2017.05.002    或    https://www.zjujournals.com/sci/CN/Y2017/V44/I5/511

[1] WEIR T.Pre-invex functions in multiobjective optimization[J]. Journal of Mathematical Analysis and Applications,1998,136(1):29-38.
[2] MOHAN S R,NEOGY S K.On invex sets and pre-invex functions[J]. Journal of Mathematical Analysis and Applications,1995,189(3):901-908.
[3] NOOR M A.Variational-like inequalities[J]. Optimization,1994,30(4):323-330.
[4] YANG X M,LI D.On properties of pre-invex functions[J]. Journal of Mathematical Analysis and Applications,2001,256(1):229-241.
[5] WANG S G,WU M X,JIA Z Z. The Matrix Inequalities[M].2nd ed. Beijing:Science Press,2006.
[6] BELLMAN R. Introduction to Matrix Analysis[M].NewYork:Mcgraw-Hill Book Company,1970.
[7] ZENG C N,XU W X,ZHOU J Z.Several notes on Hadamard theorem[J]. Journal of Math,2010,30(1):152-156.
[8] ZHANG X D,YANG S J.A note on Hadamard's inequality[J]. Acta Mathematicae Applicatae Sinca,1997,20(2):269-274.
[9] LI X Y,LENG G S.Inverse forms of Hadamard inequality and Szasz inequality[J]. Journal of Natural Science of Hunan Normal University,2007,30(2):19-21.
[10] ZHANG X D. Matrix Analysis and Applications[M].Beijing:Tsinghua University Press,2004.
[11] HORN R A,JOHNSON C R. Matrix Analysis[M].Cambridge:Cambridge University Press,1985.
[12] GOLUB G H,VAN LOAN C F. Matrix Computations[M].3rd ed.Baltimore,MD:Johns Hopkins University Press,1996.
[13] YIN H C,ZHANG H M.Eigenvalues of a class of matrices related to the positive definite matrices[J]. Journal of Zhejiang University:Science Edition,2014,41(1):1-5.
[14] ZHANG H M,DING F.A property of the eigenvalues of the symmetric positive definite matrix and the iterative algorithm for coupled Sylvester matrix equations[J]. Journal of the Franklin Institute,2014,351(1):340-357.
[15] GOVER E,KRIKORIAN N.Determinants and the volumes of parallelotopes and zonotopes[J]. Linear Algebra and Its Applications,2010,433(1):28-40.
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