Nekrasov矩阵的推广及等价条件
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摘要
广义严格对角占优矩阵是在矩阵理论和实际应用中具有重要作用和意义的一类矩阵.它在数值代数、控制论、电力系统理论、经济数学、统计学等众多领域中有着广泛的应用,受到了许多数学工作者的关注.本文我们主要考虑一类特殊矩阵,即Nekrasov矩阵. Nekrasov矩阵是广义严格对角占优矩阵的一种特殊情形,国内外学者已对Nekrasov矩阵的非奇异性、行列式及迭代矩阵的谱半径等不同方面做了大量的研究,并取得了许多重要的结果.
本文在一些近期文献的基础上,对Nekrasov的性质进行了研究,改进和推广了Nekrasov矩阵的一些结论.
第一章阐述了广义严格对角占优矩阵的理论意义和应用背景,介绍了Nekr-asov矩阵目前所取得的一些重要研究成果,给出了本文即将用到的一些定义和结论.
第二章首先指出近期关于广义Nekrasov矩阵研究中几个结果的不足,进一步,通过构造特殊矩阵和特殊向量,利用Nekrasov矩阵子阵的性质和相关不等式的放缩等技巧,对以上几个结果进行了修正,获得了一些广义Nekrasov矩阵的等价条件,并且通过实例说明了这些等价条件的有效性.
第三章推广了Nekrasov矩阵,得到了几个新的矩阵类― k层Nekrasov矩阵, k层弱Nekrasov矩阵,广义k层Nekrasov矩阵,并且通过构造特殊的正对角矩阵以及利用矩阵的性质、定义和相关不等式的放缩等技巧,给出了这几类矩阵与广义严格对角占优阵之间的关系,最后通过实例说明其有效性.
第四章对本文进行了总结,且给出了作者的研究展望.
Generalized strictly diagonally dominant matrix is a class of matrix whichplays an important role in matrix theory and real application. It has numerousapplications in many areas such as numerical algebra, control theory, electricalsystem theory, economical mathematics and statistic. Many scolars pay atten-tions to it. In this paper, we research mainly a kind of special matrices, i.e.,Nekrasov matrix. Nekrasov matrix is a special case of generalized strictly diago-nally dominant matrix, many scholars have researched many aspects of Nekrasovmatrix such as the nonsingularity, determinant and spectral radius for iterationmatrix and they have obtained lots of important results.
In this paper, based on some recent documents, we research the proper-ties about Nekrasov matrix, improving and generalizing some conclusions aboutNekerasov matrix.
In chapter one, we introduce the importance of generalized strictly diago-nally dominant matrix in theory and applied background. We detail the recentlyimportant results about Nekarasov matrix and recall some defines and conclusionsused in this paper.
In chapter two,we first point out some scarceness of some results in recentresearch about Nekrasov matrix. Further, by constructing special matrices andspecial vectores, using the properties of the submatrix of Nekrasov matrix andsome inequalities techniques, we revise this improper conclusions and obtain someequivalent conditions of the generalized Nekrasov matrix. Finally, advantages ofour results are illustrated by a numerical example.
In chapter three, we extend the Nekrasov matrix by giving some new kinds ofmatrix, which are the k-layer Nekrasov matrix, the generalized k-layer Nekrasovmatrix and the k-layer weak Nekrasov matrix. Further, by constructing specialpositive diagonal matrices, using the properties and definitions of matrix and someinequalities techniques, we obtain the relationship between these kinds of newmatrix and generalized strictly diaganally dominant matrix. Finally, advantagesof these results are illustrated by a numerical example.
In chapter four, we summary all of this paper, and give the author’s expec-tation.
本文在一些近期文献的基础上,对Nekrasov的性质进行了研究,改进和推广了Nekrasov矩阵的一些结论.
第一章阐述了广义严格对角占优矩阵的理论意义和应用背景,介绍了Nekr-asov矩阵目前所取得的一些重要研究成果,给出了本文即将用到的一些定义和结论.
第二章首先指出近期关于广义Nekrasov矩阵研究中几个结果的不足,进一步,通过构造特殊矩阵和特殊向量,利用Nekrasov矩阵子阵的性质和相关不等式的放缩等技巧,对以上几个结果进行了修正,获得了一些广义Nekrasov矩阵的等价条件,并且通过实例说明了这些等价条件的有效性.
第三章推广了Nekrasov矩阵,得到了几个新的矩阵类― k层Nekrasov矩阵, k层弱Nekrasov矩阵,广义k层Nekrasov矩阵,并且通过构造特殊的正对角矩阵以及利用矩阵的性质、定义和相关不等式的放缩等技巧,给出了这几类矩阵与广义严格对角占优阵之间的关系,最后通过实例说明其有效性.
第四章对本文进行了总结,且给出了作者的研究展望.
Generalized strictly diagonally dominant matrix is a class of matrix whichplays an important role in matrix theory and real application. It has numerousapplications in many areas such as numerical algebra, control theory, electricalsystem theory, economical mathematics and statistic. Many scolars pay atten-tions to it. In this paper, we research mainly a kind of special matrices, i.e.,Nekrasov matrix. Nekrasov matrix is a special case of generalized strictly diago-nally dominant matrix, many scholars have researched many aspects of Nekrasovmatrix such as the nonsingularity, determinant and spectral radius for iterationmatrix and they have obtained lots of important results.
In this paper, based on some recent documents, we research the proper-ties about Nekrasov matrix, improving and generalizing some conclusions aboutNekerasov matrix.
In chapter one, we introduce the importance of generalized strictly diago-nally dominant matrix in theory and applied background. We detail the recentlyimportant results about Nekarasov matrix and recall some defines and conclusionsused in this paper.
In chapter two,we first point out some scarceness of some results in recentresearch about Nekrasov matrix. Further, by constructing special matrices andspecial vectores, using the properties of the submatrix of Nekrasov matrix andsome inequalities techniques, we revise this improper conclusions and obtain someequivalent conditions of the generalized Nekrasov matrix. Finally, advantages ofour results are illustrated by a numerical example.
In chapter three, we extend the Nekrasov matrix by giving some new kinds ofmatrix, which are the k-layer Nekrasov matrix, the generalized k-layer Nekrasovmatrix and the k-layer weak Nekrasov matrix. Further, by constructing specialpositive diagonal matrices, using the properties and definitions of matrix and someinequalities techniques, we obtain the relationship between these kinds of newmatrix and generalized strictly diaganally dominant matrix. Finally, advantagesof these results are illustrated by a numerical example.
In chapter four, we summary all of this paper, and give the author’s expec-tation.
引文
[1]黄廷祝.非奇H-矩阵的简捷判据[J].计算数学,1993,3:318-328.
[2]王川龙,王华,王效俐.H-矩阵的充分条件[J].工程数学学报,2000,2:121-124.
[3]孙玉祥.非奇H-矩阵的简捷判据[J].新疆大学学报(自然科学报),1997,8:29-34.
[4]郭希娟,高益明.广义严格对角占优矩阵与非奇异M-矩阵的判定[J].高等数学计算数学学报,1999,2:23-29.
[5]高益明.矩阵广义对角占优和非奇判定[J].东北师范大学学报(自然科学报),1982,3:23-28.
[6] M.Neumann.A note on generalization of strict diagonal dominance for real ma-trices[J].Linear Algebra Appl.,1979,26:3-14.
[7] R.S.Varga.On recurring theorems on diagonal dominance[J].Linear AlgebraAppl.,1976,13:1-9.
[8]沈光星.非奇H-矩阵的新判据[J].工程数学学报,1998,4:21-27.
[9]孙玉祥.非奇H-矩阵的判定[J].工程数学学报,2000,4:45-49.
[10]干泰彬,黄廷祝.非奇H-矩阵的实用充分条件[J].计算数学,2004,2:109-116.
[11]高益明.矩阵广义对角占优和非奇的判定(II)[J].工程数学学报,1988,3:12-17.
[12]高中喜,黄廷祝,王广彬.非奇H-矩阵的充分条件[J].数学物理学报,2005,3:409-413.
[13] Y.Gao,X.Wang.Criteria for generalized diagonally dominant matrices and M-matrices[J].Linear Algebra Appl.,1992,169:257-268.
[14]逢明贤.广义对角占优矩阵的判定及应用[J].数学年刊,1985,6A(3):323-330.
[15] T.Huang.A note on generalized diagonally dominant matrices[J].Linear AlgebraAppl.,1995,225:237-242.
[16] T.Gan,T. Huang.Simple criteria for nonsingular H-matrices[J].Linear AlgebraAppl.,2003,374:317-326.
[17]李继成,黄廷祝,雷光耀.H-矩阵的实用判定[J].应用数学学报,2003,26(3):413-418.
[18]黎稳.关于广义对角占优矩阵判别的注记[J].高等学校计算数学学报,1997,1:93-96.
[19]高益明.广义对角占优矩阵与M-矩阵的判定准则[J].高等学校计算数学学报,1992,14(3):233-239.
[20] Y.Gao,X.Wang.Criteria for generalized diagonally dominant matrices and M-matrices(Ⅱ)[J].Linear Algebra Appl.,1996,248:339-353.
[21]孙玉祥,李宝贵.非奇异H-矩阵的充分条件[J].北华大学学报,2002,3:196-198.
[22]徐仲,陆全.判定广义严格对角占优矩阵的一组充分条件[J].工程数学学报,2001,18(3):11-15.
[23]李庆春,胡文杰.广义严格对角占优矩阵的判定准则[J].高校应用数学学报,1998,14A(2):229-233.
[24] Y.Sun.An improvement on a theorem by Ostrowski and its applications[J]. North-eastern Mathematical Journal,1991,4:497-502.
[25]李庆春.关于非奇H-矩阵判别条件的注记[J].北华大学学报,2004,5:393-394.
[26]李继成,张文修.H-矩阵的判定[J].高等学校计算数学学报,1999,21(3):264-268.
[27]李庆春.广义严格对角占优矩阵的判定[J].高等学校计算数学学报,1999,21(1):87-92.
[28]孙玉祥.广义对角占优矩阵的充分条件[J].高等学校计算数学学报,1997,19(3):216-223.
[29]高福顺,孙玉祥.M-矩阵的判定[J].应用数学学报,1998,21(4):535-538.
[30]王广彬,洪掁杰,朱汉锋.非奇H-矩阵的充分条件[J].上海大学学报,2003,4:358-361.
[31]黄廷祝.Ostrowski定理的推广与非奇H-矩阵的条件[J].计算数学,1994,16(1):19-24.
[32]宋乾坤.广义严格对角占优矩阵与M-矩阵的充分判据[J].高等学校计算数学学报,2004,26(4):298-305.
[33]高益明,邵新慧,刘晓艳.论几类广义严格对角占优矩阵[J].高等学校计算数学学报,1998,4:329-335.
[34]孙玉祥,吕洪斌.广义严格对角占优矩阵与非奇M-矩阵的判定[J].厦门大学学报,2001,40(5):1011-1016.
[35]王广彬,洪振杰.非奇异H-矩阵的充分条件[J].高等学校计算数学学报,2003,25(2):184-192.
[36]逄明贤,孙玉祥.M-矩阵的等价表征[J].应用数学,1995,8(1):44-50.
[37]郭晞娟,高益明.广义对角占优矩阵和M-矩阵的判定[J].数学研究与评论,1999,19(4):733-737.
[38]黎稳.广义对角占优矩阵的判别准则[J].应用数学与计算数学学报,1995,9(2):35-38.
[39]刘建州,徐映红,廖安平.广义块对角占优矩阵的判定[J].高等学校计算数学学报,2005,27(3):250-257.
[40]陈焯荣,黎稳.迭代矩阵谱半径的上界估计[J].数学物理学报,2001,21A1:8-13.
[41] D.W.Bailey,D.E.Crabtree.Bound for determinants[J].Linear Algebra Ap-ple.,1969,2:303-309.
[42]黄廷祝,徐成贤.Nekrasov矩阵的Bailey-Crabtree行列式界的注记[J].西安交通大学学报,2002,36(12).
[43] T.Szule.Some remarks on a theorem of Gudkov[J].Linear Algebra Ap-ple.,1995,225:221-235.
[44] W.Li.On Nekrasov matrices[J].Linear Algebra Apple.,1998,281:87-96.
[45]黎稳,黄廷祝.关于非奇异性判别的Gudkov定理[J].应用数学学报,1999,22(2):199-203.
[46] M.Pang,Z.Li.Generalized Nekrasov matrices and applications[J].Journal ofComputational Mathematic,2003,21(2):183-188.
[47]廖安平,刘建州.矩阵论[M].长沙:湖南大学出版社,2005.
[48]杨尚骏,卢业广,杜吉佩(译编).非负矩阵[M].山东:辽宁教育出版社,1991.
[49] R.L.Smith.On characterizing Z-matrices[J].Lin.Alg.Appl.,2001,238:99-104.
[50]李乔.矩阵论八讲[M].上海:上海科学技术出版社,1988.
[2]王川龙,王华,王效俐.H-矩阵的充分条件[J].工程数学学报,2000,2:121-124.
[3]孙玉祥.非奇H-矩阵的简捷判据[J].新疆大学学报(自然科学报),1997,8:29-34.
[4]郭希娟,高益明.广义严格对角占优矩阵与非奇异M-矩阵的判定[J].高等数学计算数学学报,1999,2:23-29.
[5]高益明.矩阵广义对角占优和非奇判定[J].东北师范大学学报(自然科学报),1982,3:23-28.
[6] M.Neumann.A note on generalization of strict diagonal dominance for real ma-trices[J].Linear Algebra Appl.,1979,26:3-14.
[7] R.S.Varga.On recurring theorems on diagonal dominance[J].Linear AlgebraAppl.,1976,13:1-9.
[8]沈光星.非奇H-矩阵的新判据[J].工程数学学报,1998,4:21-27.
[9]孙玉祥.非奇H-矩阵的判定[J].工程数学学报,2000,4:45-49.
[10]干泰彬,黄廷祝.非奇H-矩阵的实用充分条件[J].计算数学,2004,2:109-116.
[11]高益明.矩阵广义对角占优和非奇的判定(II)[J].工程数学学报,1988,3:12-17.
[12]高中喜,黄廷祝,王广彬.非奇H-矩阵的充分条件[J].数学物理学报,2005,3:409-413.
[13] Y.Gao,X.Wang.Criteria for generalized diagonally dominant matrices and M-matrices[J].Linear Algebra Appl.,1992,169:257-268.
[14]逢明贤.广义对角占优矩阵的判定及应用[J].数学年刊,1985,6A(3):323-330.
[15] T.Huang.A note on generalized diagonally dominant matrices[J].Linear AlgebraAppl.,1995,225:237-242.
[16] T.Gan,T. Huang.Simple criteria for nonsingular H-matrices[J].Linear AlgebraAppl.,2003,374:317-326.
[17]李继成,黄廷祝,雷光耀.H-矩阵的实用判定[J].应用数学学报,2003,26(3):413-418.
[18]黎稳.关于广义对角占优矩阵判别的注记[J].高等学校计算数学学报,1997,1:93-96.
[19]高益明.广义对角占优矩阵与M-矩阵的判定准则[J].高等学校计算数学学报,1992,14(3):233-239.
[20] Y.Gao,X.Wang.Criteria for generalized diagonally dominant matrices and M-matrices(Ⅱ)[J].Linear Algebra Appl.,1996,248:339-353.
[21]孙玉祥,李宝贵.非奇异H-矩阵的充分条件[J].北华大学学报,2002,3:196-198.
[22]徐仲,陆全.判定广义严格对角占优矩阵的一组充分条件[J].工程数学学报,2001,18(3):11-15.
[23]李庆春,胡文杰.广义严格对角占优矩阵的判定准则[J].高校应用数学学报,1998,14A(2):229-233.
[24] Y.Sun.An improvement on a theorem by Ostrowski and its applications[J]. North-eastern Mathematical Journal,1991,4:497-502.
[25]李庆春.关于非奇H-矩阵判别条件的注记[J].北华大学学报,2004,5:393-394.
[26]李继成,张文修.H-矩阵的判定[J].高等学校计算数学学报,1999,21(3):264-268.
[27]李庆春.广义严格对角占优矩阵的判定[J].高等学校计算数学学报,1999,21(1):87-92.
[28]孙玉祥.广义对角占优矩阵的充分条件[J].高等学校计算数学学报,1997,19(3):216-223.
[29]高福顺,孙玉祥.M-矩阵的判定[J].应用数学学报,1998,21(4):535-538.
[30]王广彬,洪掁杰,朱汉锋.非奇H-矩阵的充分条件[J].上海大学学报,2003,4:358-361.
[31]黄廷祝.Ostrowski定理的推广与非奇H-矩阵的条件[J].计算数学,1994,16(1):19-24.
[32]宋乾坤.广义严格对角占优矩阵与M-矩阵的充分判据[J].高等学校计算数学学报,2004,26(4):298-305.
[33]高益明,邵新慧,刘晓艳.论几类广义严格对角占优矩阵[J].高等学校计算数学学报,1998,4:329-335.
[34]孙玉祥,吕洪斌.广义严格对角占优矩阵与非奇M-矩阵的判定[J].厦门大学学报,2001,40(5):1011-1016.
[35]王广彬,洪振杰.非奇异H-矩阵的充分条件[J].高等学校计算数学学报,2003,25(2):184-192.
[36]逄明贤,孙玉祥.M-矩阵的等价表征[J].应用数学,1995,8(1):44-50.
[37]郭晞娟,高益明.广义对角占优矩阵和M-矩阵的判定[J].数学研究与评论,1999,19(4):733-737.
[38]黎稳.广义对角占优矩阵的判别准则[J].应用数学与计算数学学报,1995,9(2):35-38.
[39]刘建州,徐映红,廖安平.广义块对角占优矩阵的判定[J].高等学校计算数学学报,2005,27(3):250-257.
[40]陈焯荣,黎稳.迭代矩阵谱半径的上界估计[J].数学物理学报,2001,21A1:8-13.
[41] D.W.Bailey,D.E.Crabtree.Bound for determinants[J].Linear Algebra Ap-ple.,1969,2:303-309.
[42]黄廷祝,徐成贤.Nekrasov矩阵的Bailey-Crabtree行列式界的注记[J].西安交通大学学报,2002,36(12).
[43] T.Szule.Some remarks on a theorem of Gudkov[J].Linear Algebra Ap-ple.,1995,225:221-235.
[44] W.Li.On Nekrasov matrices[J].Linear Algebra Apple.,1998,281:87-96.
[45]黎稳,黄廷祝.关于非奇异性判别的Gudkov定理[J].应用数学学报,1999,22(2):199-203.
[46] M.Pang,Z.Li.Generalized Nekrasov matrices and applications[J].Journal ofComputational Mathematic,2003,21(2):183-188.
[47]廖安平,刘建州.矩阵论[M].长沙:湖南大学出版社,2005.
[48]杨尚骏,卢业广,杜吉佩(译编).非负矩阵[M].山东:辽宁教育出版社,1991.
[49] R.L.Smith.On characterizing Z-matrices[J].Lin.Alg.Appl.,2001,238:99-104.
[50]李乔.矩阵论八讲[M].上海:上海科学技术出版社,1988.