几类特殊矩阵的幂与乘积
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摘要
在第一章，我们给出全文涉及到的一些基本概念及结论．在第二章，我们给出邵嘉裕教授关于非负矩阵可以分解成不可约非负阵乘积的充要条件的结果的新证明，并且确定最少的因子个数．在第三章，我们研究了给定秩的0-1矩阵的正整数次幂中1的可能个数．在第四章，我们研究了幂仍为0-1矩阵的0-1矩阵中1的可能个数．在第五章，我们研究了Toeplitz矩阵和Hankel矩阵的幂，给出了Cauchy-Toeplitz矩阵和Cauchy—Hankel矩阵的Frobenius范数和谱范数的下界．在第六章，我们给出了计算一种偶阶反三对角矩阵的任意次幂的显式表达式．本文讨论如下五类问题．
1．非负矩阵的分解
矩阵分解问题在矩阵论和矩阵计算中都十分重要，这类问题讨论的是将一个矩阵分解成若干个特定类型矩阵的乘积或者和的问题．在1985年，邵嘉裕教授给出了一个非负方阵可以分解成不可约非负矩阵的乘积的充要条件([38])．本文用图论方法重新证明了他的定理．并且证明了，若一个非负方阵可以分解成不可约非负矩阵的乘积，则可以要求因子的个数至多为3．所用的证明方法是构造性的，可以具体写出各个因子矩阵．
2．给定秩的0-1矩阵的幂
0-1矩阵是元素取值0或1的非负矩阵，这样的矩阵经常出现在组合学和图论中．在2005年，胡奇、李娅琴和詹兴致确定了给定秩的一般0-1矩阵和给定秩的对称0-1矩阵中1的可能个数([24])．随后詹兴致教授又提出，在给定秩的0-1矩阵的某个正整数次幂中，1的可能个数又是哪些呢?本文部分地回答了此问题，
3．幂仍为0-1矩阵的0-1矩阵
直觉上，如果一个0-1矩阵含有太多的元素等于1，那么它的幂不太可能仍是0-1矩阵．在2007年詹兴致教授在讨论班上提出下面的问题：
给定正整数n和k，如果n阶0-1矩阵A的k次幂仍为0-1矩阵，那么A中1的最多个数是多少呢?进一步地，如何来刻画这些1的个数取到最大值的0-1矩阵呢?本文解决了詹的问题中k=2的情况．具体来说，我们确定了平方仍是0-1矩阵的0-1矩阵中1的最大个数，同时刻画了取到最大值的0-1矩阵．并且，我们初步研究了k＞2的情况．
4．特殊矩阵的幂与范数
在矩阵论中有一些结构特殊并且在应用中非常重要的矩阵．
一方面，本文研究了Toeplitz矩阵和Hankel矩阵的幂．Tamir Shalom给出了使得Toeplitz矩阵的任意次幂仍为Toeplitz矩阵的充要条件([37])．我们给出了他的结论的一个新证明．另外，我们给出了使得Hankel矩阵的任意次幂仍是Hankel矩阵的一个充分条件．
另一方面，本文利用多伽玛函数及辅助矩阵给出了一般Cauchy-Toeplitz矩阵和一般Cauchy-Hankel矩阵的Frobenius范数和谱范数的下界．
5．一种反三对角矩阵的任意次幂
在差分方程和微分方程求解中有时需要计算三对角矩阵和反三对角矩阵的任意次幂([1]，[25]，[33])．本文给出了计算一种偶阶反三对角矩阵的任意次幂的显式表达式．
In the first chapter,we give some basic concepts and conclusions involvedin this dissertation.In the second chapter,we give a new proof of ProfessorJiayu Shao's result on a necessary and sufficient condition for a nonnegativematrix to be decomposed into a product of irreducible nonnegative matrices,and determine the least possible number of factors.In the third chapter,wediscuss the possible numbers of ones in the powers of 0-1 matrices with agiven rank.In the fourth chapter,we discuss the possible numbers of onesin 0-1 matrices whose powers are still 0-1 matrices.In the fifth chapter,westudy the powers of Toeplitz matrices and Hankel matrices,and give lowerbounds for Frobenius norms and spectral norms of Cauchy-Toeplitz matricesand Cauchy-Hankel matrices respectively.In the sixth chapter,we give anexplicit expression for any power of a certain type of anti-tridiagonal matricesof even order.This dissertation discusses five kinds of problems as follows.
1.Decomposition of nonnegative matrices
Matrix decomposition problems are important in both matrix theory andmatrix computations.This kind of problems discusses the problems of decom-posing a matrix into products or sums of several matrices of some special kinds.Professor Jiayu Shao gave a necessary and sufficient condition for a nonnegativematrix to be decomposed into a product of irreducible nonnegative matricesin 1985 ([38]).We reprove his result using a graph-theoretic method,and alsoshow that when such a decomposition is possible,the number of factors canbe required to be at most three.The methods used here are constructive,and they give an algorithm to produce the factors.
2.Powers of 0-1 matrices with a given rank
A 0-1 matrix is a matrix whose entries are 0 or 1.Such matrices arisefrequently in combinatorics and graph theory.In 2005,Qi Hu,Yaqin Li andXingzhi Zhan determined the possible numbers of ones in a 0-1 matrix witha given rank in the generic case and in the symmetric case ([24]).ProfessorXingzhi Zhan asked later:what are the possible numbers of ones in the powersof a 0-1 matrix with a given rank? We answer this question partially.
3.0-1 matrices whose powers are still 0-1 matrices
Intuitively,if a 0-1 matrix has too many ones,its powers can not be still0-1 matrices.In 2007 Professor Xingzhi Zhan posed the following problem ata seminar:
Given positive integers n and k,if the kth power of a 0-1 matrix A oforder n is still a 0-1 matrix,then what is the maximum number of ones in A?Furthermore,how to characterize those matrices which attain the maximumnumber? We solve the case k=2 of Zhan's problem.That is,we determinethe maximum number of ones in the 0-1 matrices whose squares are still 0-1matrices,and the maximizing matrices are also specified.Furthermore,westudy the case k＞2 partially.
4.Powers and norms of special matrices
There are some matrices with special structures which have importantapplications.
On the one hand,we discuss the powers of Toeplitz matrices and Han-kel matrices.Tamir Shalom gave a necessary and sufficient condition for thepowers of Toeplitz matrices to be still Toeplitz matrices ([37]).We give a newproof of his result.Moreover,we give a sufficient condition for the powers ofHankel matrices to be still Hankel matrices.
On the other hand,we give lower bounds for the Frobenius norms andspectral norms of Cauchy-Toeplitz matrices and Cauchy-Hankel matrices, using the polygamma function and assistant matrices.
5.Powers of one kind of anti-tridiagonal matrices
In solving some difference equations and differential equations we need tocompute the arbitrary powers of square matrices ([1],[25],[33]).We derive anexplicit expression for the powers of some kind of anti-tridiagonal matrices ofeven order.
1．非负矩阵的分解
矩阵分解问题在矩阵论和矩阵计算中都十分重要，这类问题讨论的是将一个矩阵分解成若干个特定类型矩阵的乘积或者和的问题．在1985年，邵嘉裕教授给出了一个非负方阵可以分解成不可约非负矩阵的乘积的充要条件([38])．本文用图论方法重新证明了他的定理．并且证明了，若一个非负方阵可以分解成不可约非负矩阵的乘积，则可以要求因子的个数至多为3．所用的证明方法是构造性的，可以具体写出各个因子矩阵．
2．给定秩的0-1矩阵的幂
0-1矩阵是元素取值0或1的非负矩阵，这样的矩阵经常出现在组合学和图论中．在2005年，胡奇、李娅琴和詹兴致确定了给定秩的一般0-1矩阵和给定秩的对称0-1矩阵中1的可能个数([24])．随后詹兴致教授又提出，在给定秩的0-1矩阵的某个正整数次幂中，1的可能个数又是哪些呢?本文部分地回答了此问题，
3．幂仍为0-1矩阵的0-1矩阵
直觉上，如果一个0-1矩阵含有太多的元素等于1，那么它的幂不太可能仍是0-1矩阵．在2007年詹兴致教授在讨论班上提出下面的问题：
给定正整数n和k，如果n阶0-1矩阵A的k次幂仍为0-1矩阵，那么A中1的最多个数是多少呢?进一步地，如何来刻画这些1的个数取到最大值的0-1矩阵呢?本文解决了詹的问题中k=2的情况．具体来说，我们确定了平方仍是0-1矩阵的0-1矩阵中1的最大个数，同时刻画了取到最大值的0-1矩阵．并且，我们初步研究了k＞2的情况．
4．特殊矩阵的幂与范数
在矩阵论中有一些结构特殊并且在应用中非常重要的矩阵．
一方面，本文研究了Toeplitz矩阵和Hankel矩阵的幂．Tamir Shalom给出了使得Toeplitz矩阵的任意次幂仍为Toeplitz矩阵的充要条件([37])．我们给出了他的结论的一个新证明．另外，我们给出了使得Hankel矩阵的任意次幂仍是Hankel矩阵的一个充分条件．
另一方面，本文利用多伽玛函数及辅助矩阵给出了一般Cauchy-Toeplitz矩阵和一般Cauchy-Hankel矩阵的Frobenius范数和谱范数的下界．
5．一种反三对角矩阵的任意次幂
在差分方程和微分方程求解中有时需要计算三对角矩阵和反三对角矩阵的任意次幂([1]，[25]，[33])．本文给出了计算一种偶阶反三对角矩阵的任意次幂的显式表达式．
In the first chapter,we give some basic concepts and conclusions involvedin this dissertation.In the second chapter,we give a new proof of ProfessorJiayu Shao's result on a necessary and sufficient condition for a nonnegativematrix to be decomposed into a product of irreducible nonnegative matrices,and determine the least possible number of factors.In the third chapter,wediscuss the possible numbers of ones in the powers of 0-1 matrices with agiven rank.In the fourth chapter,we discuss the possible numbers of onesin 0-1 matrices whose powers are still 0-1 matrices.In the fifth chapter,westudy the powers of Toeplitz matrices and Hankel matrices,and give lowerbounds for Frobenius norms and spectral norms of Cauchy-Toeplitz matricesand Cauchy-Hankel matrices respectively.In the sixth chapter,we give anexplicit expression for any power of a certain type of anti-tridiagonal matricesof even order.This dissertation discusses five kinds of problems as follows.
1.Decomposition of nonnegative matrices
Matrix decomposition problems are important in both matrix theory andmatrix computations.This kind of problems discusses the problems of decom-posing a matrix into products or sums of several matrices of some special kinds.Professor Jiayu Shao gave a necessary and sufficient condition for a nonnegativematrix to be decomposed into a product of irreducible nonnegative matricesin 1985 ([38]).We reprove his result using a graph-theoretic method,and alsoshow that when such a decomposition is possible,the number of factors canbe required to be at most three.The methods used here are constructive,and they give an algorithm to produce the factors.
2.Powers of 0-1 matrices with a given rank
A 0-1 matrix is a matrix whose entries are 0 or 1.Such matrices arisefrequently in combinatorics and graph theory.In 2005,Qi Hu,Yaqin Li andXingzhi Zhan determined the possible numbers of ones in a 0-1 matrix witha given rank in the generic case and in the symmetric case ([24]).ProfessorXingzhi Zhan asked later:what are the possible numbers of ones in the powersof a 0-1 matrix with a given rank? We answer this question partially.
3.0-1 matrices whose powers are still 0-1 matrices
Intuitively,if a 0-1 matrix has too many ones,its powers can not be still0-1 matrices.In 2007 Professor Xingzhi Zhan posed the following problem ata seminar:
Given positive integers n and k,if the kth power of a 0-1 matrix A oforder n is still a 0-1 matrix,then what is the maximum number of ones in A?Furthermore,how to characterize those matrices which attain the maximumnumber? We solve the case k=2 of Zhan's problem.That is,we determinethe maximum number of ones in the 0-1 matrices whose squares are still 0-1matrices,and the maximizing matrices are also specified.Furthermore,westudy the case k＞2 partially.
4.Powers and norms of special matrices
There are some matrices with special structures which have importantapplications.
On the one hand,we discuss the powers of Toeplitz matrices and Han-kel matrices.Tamir Shalom gave a necessary and sufficient condition for thepowers of Toeplitz matrices to be still Toeplitz matrices ([37]).We give a newproof of his result.Moreover,we give a sufficient condition for the powers ofHankel matrices to be still Hankel matrices.
On the other hand,we give lower bounds for the Frobenius norms andspectral norms of Cauchy-Toeplitz matrices and Cauchy-Hankel matrices, using the polygamma function and assistant matrices.
5.Powers of one kind of anti-tridiagonal matrices
In solving some difference equations and differential equations we need tocompute the arbitrary powers of square matrices ([1],[25],[33]).We derive anexplicit expression for the powers of some kind of anti-tridiagonal matrices ofeven order.
引文
[1]R.P.Agarwal,Difference Equations and Inequalities,Marcel Dekker,1992.
[2]R.B.Bapat,T.E.S.Raghavan,Nonnegative Matrices and Applications,Cambridge University Press,Cambridge,1997.
[3]C.Berge,Graph and Hypergraphs,North-Holland,1976.
[4]R.Bhatia,Matrix Analysis,GTM 169,Springer-Verlag,New York,1997.
[5]B.Bollobas,Graph Theory,Spring-Verlag,New York,1979.
[6]D.Bozkurt,On the bounds for the l_p norm of almost Cauchy-Toeplitz matrix,Turkish Journal of Mathematics,20(1996),545-552.
[7]D.Bozkurt,On the l_p norms of Cauchy-Toeplitz matrices,Linear and Multilinear Algebra,44(1998),341-346.
[8]R.A.Brualdi,Introductory Combinatorics,Pearson Education Asia Limited Press,2004.
[9]R.A.Brualdi,Matrices permutation equivalent to irreducible matrices and applications,Linear and Multilinear Algebra,7(1979),1-12.
[10]R.A.Brualdi,M.Lewin,On powers of nonnegative matrices,Linear Algebra and its Applications,43(1982),87-97.
[11]R.A.Brualdi,H.J.Ryser,Combinatorial Matrix Theory,Cambridge University Press,Cambridge,1991.
[12]H.Chen,M.Chen,On products of three triangular matrices over associative rings,Linear Algebra and its Applications,387(2004),297-311.
[13]陈景良，陈向晖，特殊矩阵，清华大学出版社，北京，2000．
[14]J.E.Cohen,U.G.Rothblum,Nonnegative ranks,decompositions,and factorizations of nonnegative matrices,Linear Algebra and its Applications,190(1993),149-168.
[15]L.Euler,De progressionibus transcendentibus seu quarum termini generales algebraice dari nequeunt,Comm.Acad.Sci.Petropolitanae 5(1730),36-57.
[16]S.Furtado,L.Iglesias,F.C.Silva,Products of matrices with prescribed spectra and ranks,Linear Algebra and its Applications,340(2002),137-147.
[17]M.J.Goldstein,Reduction of the eigenproblem for Hermite persymmetric matrices,Math.Comp.,125(1974)237-238.
[18]G.H.Golub,C.F.Van Loan,Matrix Computations,The Johns Hopkins University Press,1996.
[19]A.D.Güng(o|¨)r,Lower bounds for the norms of Cauchy-Toeplitz and Cauchy-Hankel matrices,Applied Mathematics and Computation,157(2004),599-604.
[20]R.P.Gupta,On decompositions of a multigraph with spanning subgraphs,Bull.Amer.Math.Soc.,80(1974),500-502.
[21]韩士安，林磊，近世代数，科学出版社，北京，2004．
[22]R.A.Horn,C.R.Johnson,Matrix Analysis,Cambridge University Press,1985.
[23]R.A.Horn,C.R.Johnson,Topices in Matrix Analysis,Cambridge University Press,1991.
[24]Q.Hu,Y.Li,X.Zhan,Possible numbers of ones in 0-1 matrices with a given rank,Linear and Multilinear Algebra,53(2005),435-443.
[25]G.James,Advanced Modern Engineering Mathematics,Addison-Wesley,1994.
[26]M.Lewin,On nonnegative matrices,Pacific Journal Mathematics,No.3,36(1971).
[27]李乔，矩阵论八讲，上海科学技术出版社，1988．
[28]柳柏濂，组合矩阵论，科学出版社，2005．
[29]H.Minc,Nonnegative Matrices,John Wiley & Sons,1988.
[30]R.Moenck,On computing closed forms for summations,Proceedings of MACSYMA users' conference,Berkley,1977,225-236.
[31]K.R.Nagarajan,M.P.Devasahayam,T.Soundararajan,Products of three triangular matrices,Linear Algebra and its Applications,292(1999),61-71.
[32]S.V.Parter,On the distribution of the singular values of Toeplitz matrices,Linear Algebra and its Applications,80(1986),115-130.
[33]J.Rimas,Investigation of the dynamics of mutually synchronized systems,Telecommunications and Radio Engineer 32(1977),68-79.
[34]J.Rimas,On computing of arbitrary positive integer powers for one type of symmetric anti-tridiagonal matrices of odd order,Applied Mathematics and Computation,210(2009),64-71.
[35]O.Rojo,Further bounds for the smallest singular value and the spectral condition number,Computers and Mathematics with Applications,38(1999),215-228.
[36]M.Russell,Multilinear Algebra,Gordon & Breach,Amsterdam,1997.
[37]T.Shalom,On algebras of Toeplitz matrices,Linear Algebra and its Applications,96(1987),211-226.
[38]J.Shao,Products of irreducible matrices,Linear Algebra and its Applications,68(1985),131-143.
[39]G.D.Smith,Numerical solution of partial differential equations:Finite Difference Methods,2nd edit.,Oxford University Press,London,1978.
[40]S.Solak,D.Bozkurt,On the spectral norms of Cauchy-Toeplitz and Cauchy-Hankel matrices,Applied Mathematics and Computation,140(2003),231-238.
[41]A.R.Sourour,K.Tang,Factorization of singular matrices,Proceedings of the American Mathematical Society,116(1992),629-634.
[42]孙继广，矩阵扰动分析(第二版)，科学出版社，2001．
[43]R.Turkmen,D.Bozkurt,On the bounds for the norms of Cauchy-Toeplitz and Cauchy-Hankel matrices,Applied Mathematics and Computation,132(2002),633-642.
[44]E.E.Tyrtyshnikov,Cauchy-Toeplitz matrices and some applications,Linear Algebra and its Applications,149(1991),1-18.
[45]J.Wang,P.Wu,Products of unipotent matrices of index 2,Linear Algebra and its Applications,149(1991),111-123.
[46]D.B.West,Introduction to Graph Theory,Prentice-Hall,2001.
[47]P.Wu,Products of nilpotent matrices,Linear Algebra and its Applications,96 (1987),227-232.
[48]P.Wu,Products of positive semidefinite matrices,Linear Algebra and its Applications,111(1988),53-61.
[49]Q.Yin,On computing of arbitrary positive powers for anti-tridiagonal matrices of even order,Applied Mathematics and Computation,204(2008),288-298.
[50]詹兴致，矩阵论，高等教育出版社，北京，2008．
[2]R.B.Bapat,T.E.S.Raghavan,Nonnegative Matrices and Applications,Cambridge University Press,Cambridge,1997.
[3]C.Berge,Graph and Hypergraphs,North-Holland,1976.
[4]R.Bhatia,Matrix Analysis,GTM 169,Springer-Verlag,New York,1997.
[5]B.Bollobas,Graph Theory,Spring-Verlag,New York,1979.
[6]D.Bozkurt,On the bounds for the l_p norm of almost Cauchy-Toeplitz matrix,Turkish Journal of Mathematics,20(1996),545-552.
[7]D.Bozkurt,On the l_p norms of Cauchy-Toeplitz matrices,Linear and Multilinear Algebra,44(1998),341-346.
[8]R.A.Brualdi,Introductory Combinatorics,Pearson Education Asia Limited Press,2004.
[9]R.A.Brualdi,Matrices permutation equivalent to irreducible matrices and applications,Linear and Multilinear Algebra,7(1979),1-12.
[10]R.A.Brualdi,M.Lewin,On powers of nonnegative matrices,Linear Algebra and its Applications,43(1982),87-97.
[11]R.A.Brualdi,H.J.Ryser,Combinatorial Matrix Theory,Cambridge University Press,Cambridge,1991.
[12]H.Chen,M.Chen,On products of three triangular matrices over associative rings,Linear Algebra and its Applications,387(2004),297-311.
[13]陈景良，陈向晖，特殊矩阵，清华大学出版社，北京，2000．
[14]J.E.Cohen,U.G.Rothblum,Nonnegative ranks,decompositions,and factorizations of nonnegative matrices,Linear Algebra and its Applications,190(1993),149-168.
[15]L.Euler,De progressionibus transcendentibus seu quarum termini generales algebraice dari nequeunt,Comm.Acad.Sci.Petropolitanae 5(1730),36-57.
[16]S.Furtado,L.Iglesias,F.C.Silva,Products of matrices with prescribed spectra and ranks,Linear Algebra and its Applications,340(2002),137-147.
[17]M.J.Goldstein,Reduction of the eigenproblem for Hermite persymmetric matrices,Math.Comp.,125(1974)237-238.
[18]G.H.Golub,C.F.Van Loan,Matrix Computations,The Johns Hopkins University Press,1996.
[19]A.D.Güng(o|¨)r,Lower bounds for the norms of Cauchy-Toeplitz and Cauchy-Hankel matrices,Applied Mathematics and Computation,157(2004),599-604.
[20]R.P.Gupta,On decompositions of a multigraph with spanning subgraphs,Bull.Amer.Math.Soc.,80(1974),500-502.
[21]韩士安，林磊，近世代数，科学出版社，北京，2004．
[22]R.A.Horn,C.R.Johnson,Matrix Analysis,Cambridge University Press,1985.
[23]R.A.Horn,C.R.Johnson,Topices in Matrix Analysis,Cambridge University Press,1991.
[24]Q.Hu,Y.Li,X.Zhan,Possible numbers of ones in 0-1 matrices with a given rank,Linear and Multilinear Algebra,53(2005),435-443.
[25]G.James,Advanced Modern Engineering Mathematics,Addison-Wesley,1994.
[26]M.Lewin,On nonnegative matrices,Pacific Journal Mathematics,No.3,36(1971).
[27]李乔，矩阵论八讲，上海科学技术出版社，1988．
[28]柳柏濂，组合矩阵论，科学出版社，2005．
[29]H.Minc,Nonnegative Matrices,John Wiley & Sons,1988.
[30]R.Moenck,On computing closed forms for summations,Proceedings of MACSYMA users' conference,Berkley,1977,225-236.
[31]K.R.Nagarajan,M.P.Devasahayam,T.Soundararajan,Products of three triangular matrices,Linear Algebra and its Applications,292(1999),61-71.
[32]S.V.Parter,On the distribution of the singular values of Toeplitz matrices,Linear Algebra and its Applications,80(1986),115-130.
[33]J.Rimas,Investigation of the dynamics of mutually synchronized systems,Telecommunications and Radio Engineer 32(1977),68-79.
[34]J.Rimas,On computing of arbitrary positive integer powers for one type of symmetric anti-tridiagonal matrices of odd order,Applied Mathematics and Computation,210(2009),64-71.
[35]O.Rojo,Further bounds for the smallest singular value and the spectral condition number,Computers and Mathematics with Applications,38(1999),215-228.
[36]M.Russell,Multilinear Algebra,Gordon & Breach,Amsterdam,1997.
[37]T.Shalom,On algebras of Toeplitz matrices,Linear Algebra and its Applications,96(1987),211-226.
[38]J.Shao,Products of irreducible matrices,Linear Algebra and its Applications,68(1985),131-143.
[39]G.D.Smith,Numerical solution of partial differential equations:Finite Difference Methods,2nd edit.,Oxford University Press,London,1978.
[40]S.Solak,D.Bozkurt,On the spectral norms of Cauchy-Toeplitz and Cauchy-Hankel matrices,Applied Mathematics and Computation,140(2003),231-238.
[41]A.R.Sourour,K.Tang,Factorization of singular matrices,Proceedings of the American Mathematical Society,116(1992),629-634.
[42]孙继广，矩阵扰动分析(第二版)，科学出版社，2001．
[43]R.Turkmen,D.Bozkurt,On the bounds for the norms of Cauchy-Toeplitz and Cauchy-Hankel matrices,Applied Mathematics and Computation,132(2002),633-642.
[44]E.E.Tyrtyshnikov,Cauchy-Toeplitz matrices and some applications,Linear Algebra and its Applications,149(1991),1-18.
[45]J.Wang,P.Wu,Products of unipotent matrices of index 2,Linear Algebra and its Applications,149(1991),111-123.
[46]D.B.West,Introduction to Graph Theory,Prentice-Hall,2001.
[47]P.Wu,Products of nilpotent matrices,Linear Algebra and its Applications,96 (1987),227-232.
[48]P.Wu,Products of positive semidefinite matrices,Linear Algebra and its Applications,111(1988),53-61.
[49]Q.Yin,On computing of arbitrary positive powers for anti-tridiagonal matrices of even order,Applied Mathematics and Computation,204(2008),288-298.
[50]詹兴致，矩阵论，高等教育出版社，北京，2008．