关于广义Vandermonde矩阵的若干问题
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摘要
本文围绕Vandermonde矩阵和几种广义Vandermonde矩阵(如Cauchy-Vandermonde矩阵、合流Vandermonde矩阵、函数形式的合流Vandermonde矩阵以及q-adic Vandermonde矩阵)展开讨论，归纳总结了它们的若干性质。
首先介绍了Vandermonde矩阵的概念和若干性质，如与多项式插值问题之间的联系，以及Vandermonde矩阵的三角分解等。
其次，本文针对几种广义Vandermonde矩阵进行了归纳和推导。例如，Cauchy-Vandermonde矩阵在Lagrange类型插值问题中可以看作其系数矩阵。同时本文给出了Cauchy-Vandermonde矩阵的其它性质，如逆矩阵的表示等。此外，围绕合流Vandermonde矩阵，一方面给出它的插值意义，得到合流Vandermonde矩阵与Hermite插值问题之间的联系：另一方面给出行列式值的递推公式等。对于函数形式的合流Vandermonde矩阵和q-adic Vandermonde矩阵，同样得到了它们与相关插值问题的联系以及其它性质。
最后，通过给定一个具有相同重度的结点序列，构造对应的合流Vandermonde矩阵和特殊的Hankel矩阵，给出了这两种矩阵之间的关系。此外，对应结点构造一种特殊的Toeplitz矩阵，类似可得到一系列相关性质。
In this thesis, we introduce the Vandermonde matrix and several generalizedVandermonde matrices(Cauchy-Vandermonde matrix, confluent Vandermonde matrix,functional confluent Vandermonde matrix and q-adic Vandermonde matrix), anddiscuss some important properties of these matrices.
First, we introduce the Vandermonde matrix, and give its relationship topolynomial interpolation problem. Then we discuss its other properties such astriangular decomposition.
Secondly, we sum up some generalized Vandermonde matrices, and describetheir properties. For example, we interpret Cauchy-Vandermonde matrix as thecoefficient matrix of a Lagrange interpolation problem, and give representation ofinversion. We discuss the confluent Vandermonde matrix, and give its relationship toHermite interpolation problem. Then, we give some similar properties of thefunctional confluent Vandermonde matrix and q-adic Vandermonde matrix.
Finally, we give the finite sequences of interpolation nodes in which each nodehas the same multiplicity, construct its corresponding confluent Vandermonde matrixand Hankel matrix, and prove the relationship between them. Also, we prove thesimilar properties of Toeplitz matrix.
首先介绍了Vandermonde矩阵的概念和若干性质，如与多项式插值问题之间的联系，以及Vandermonde矩阵的三角分解等。
其次，本文针对几种广义Vandermonde矩阵进行了归纳和推导。例如，Cauchy-Vandermonde矩阵在Lagrange类型插值问题中可以看作其系数矩阵。同时本文给出了Cauchy-Vandermonde矩阵的其它性质，如逆矩阵的表示等。此外，围绕合流Vandermonde矩阵，一方面给出它的插值意义，得到合流Vandermonde矩阵与Hermite插值问题之间的联系：另一方面给出行列式值的递推公式等。对于函数形式的合流Vandermonde矩阵和q-adic Vandermonde矩阵，同样得到了它们与相关插值问题的联系以及其它性质。
最后，通过给定一个具有相同重度的结点序列，构造对应的合流Vandermonde矩阵和特殊的Hankel矩阵，给出了这两种矩阵之间的关系。此外，对应结点构造一种特殊的Toeplitz矩阵，类似可得到一系列相关性质。
In this thesis, we introduce the Vandermonde matrix and several generalizedVandermonde matrices(Cauchy-Vandermonde matrix, confluent Vandermonde matrix,functional confluent Vandermonde matrix and q-adic Vandermonde matrix), anddiscuss some important properties of these matrices.
First, we introduce the Vandermonde matrix, and give its relationship topolynomial interpolation problem. Then we discuss its other properties such astriangular decomposition.
Secondly, we sum up some generalized Vandermonde matrices, and describetheir properties. For example, we interpret Cauchy-Vandermonde matrix as thecoefficient matrix of a Lagrange interpolation problem, and give representation ofinversion. We discuss the confluent Vandermonde matrix, and give its relationship toHermite interpolation problem. Then, we give some similar properties of thefunctional confluent Vandermonde matrix and q-adic Vandermonde matrix.
Finally, we give the finite sequences of interpolation nodes in which each nodehas the same multiplicity, construct its corresponding confluent Vandermonde matrixand Hankel matrix, and prove the relationship between them. Also, we prove thesimilar properties of Toeplitz matrix.
引文
[1] 程伟健，贺冬冬，范德蒙行列式在微秋分中的应用，大学数学，2004年03期．
[2] 刘建中，范德蒙行列式的一个性质的证明及其应用，河北大学学报(自然科学版)，2000年01期．
[3] 赵强，一类行列式的计算一范德蒙行列式和行列式乘积的应用，云南民族学院学报(自然科学版)，2001年03期．
[4] 黄贤通，张朝全，任金威，GF(Pn)上的插值理论及其在信息隐藏中的应用，大学数学，2005 Vol．21 No．5 P．1-6．
[5] I.Gohgerg, V.Olshevsky, Fast inversion of Chebyshev-Vandermonde matrices,Numer.Math.67 (1) (1994) 71-92.
[6] L.verde-star, Biorthogonal polynomial bases and Vandermonde-like matrices,Stud.Appl.Math.95 (1995) 269-295.
[7] G.Heinig,Generalized Cauchy-Vandermonde matrices,Linear Algebra Appl.270 (1998) 45-77.
[8] Halil Oruc and George M.Phillips, Explicit factorization of the Vandermonde matrix,Linear Algebra and its Applications.315: 113-123, 2000.
[9] M.Gasca,J.J.Martinez,G.Muhlbach,Computation of rational interpolants with prescribed poles,J.Comput.Appl.Math 26 (1989) 297-309.
[10] P.Junghanns,D.Oestreich,Numerische Losung des Staudammproblems mit Drainage,Z.Angew.Math.Mech. 69 (1989) 83-92.
[11] G.Heinig Inversion of Toeplitz Matrices via Generlized Cauchy Matrices and Rational Interpolations, vol.2,Akademie Verlag,1994, pp. 707-711.
[12] G.Heinig Inversion of generalized Cauchy Matrices and other classes of structured matrices,Linear Algebra for Signal Processing,IMA Vol.in Math.Appl.69 (1994) 95-114.
[13] G.Heinig,A.Bojanczyk,Transformation techniques for Toeplitz and Toeplitz-plus-Hankei structured matrices Ⅰ. transformations, Linear Algebra Appl.254 (1997) 193-226.
[14] J.J.Martfnez and J.M.Pena,Factorizations of Cauchy-Vandermonde matrices, Linear Algebra and its Applications. 284: 229-237, 1998.
[15] Tiio Finck,Georg Heinig and Karla Rost,An inversion Formula and Fast Algorithms for Cauchy-Vandermonde Matrices, Linear Algebra and its Applications. 183: 179-191, 1993.
[16] 王仁宏，朱功勤，有理函数逼近及其应用[M]，北京：科学出版社，2004：85-93．
[17] 王仁宏，梁学章，多元函数逼近[M]，北京：科学出版社，1988：62-69．
[18] 崔蓉蓉，黄有度，一元切触有理插值存在性的判别方法，大学学报，22(4)，2006．
[19] 马锦锦，二元切触有理插值，阜阳师范学院学报(自然科学版)，23(1)，2006．
[20] Z.Vavrin,Confluent Cauchy and Cauchy-Vandermonde Matrices,Linear Algebra and its Applications.258: 271-293, 1997.
[21] 盛中平，广义Vandermonde行列式及其应用，高等学校计算数学学报，1996年9，第3期．
[22] T.T.Ha and J.A.Gibson,A Note on the Determinant of a functional Confluent Vandermonde Matrix and Controllability, Linear Algebra and its Applications. 30: 69-75, 1980.
[23] G. chen and Z. Yang, Bezoutian Representation Via Vandermonde Matrices, Linear Algebra and its Applications. 186: 37-44, 1993.
[24] U.Helmke and P.A.Fuhrman, Bezoutians, Linear Algebra and its Applications. 122-124: 1039-1097, 1989.
[25] G.Sansigre and M.Alvarez,On Bezoutian Reduction with Vandermonde Matrices, Linear Algebra and its Applications. 121: 401-408, 1989.
[26] 杨正宏，胡永建和陈公宁，q-adic范德蒙型矩阵的位移结构与逆公式，北京师范大学学报(自然科学版)，39(2)，2003．
[27] V.Mani and R.Hartwig, Some Properties of the Q-adic Vandermonde Matrix,the Electronic Journal of Linear Algebra.1: 18-33, 1996.
[2] 刘建中，范德蒙行列式的一个性质的证明及其应用，河北大学学报(自然科学版)，2000年01期．
[3] 赵强，一类行列式的计算一范德蒙行列式和行列式乘积的应用，云南民族学院学报(自然科学版)，2001年03期．
[4] 黄贤通，张朝全，任金威，GF(Pn)上的插值理论及其在信息隐藏中的应用，大学数学，2005 Vol．21 No．5 P．1-6．
[5] I.Gohgerg, V.Olshevsky, Fast inversion of Chebyshev-Vandermonde matrices,Numer.Math.67 (1) (1994) 71-92.
[6] L.verde-star, Biorthogonal polynomial bases and Vandermonde-like matrices,Stud.Appl.Math.95 (1995) 269-295.
[7] G.Heinig,Generalized Cauchy-Vandermonde matrices,Linear Algebra Appl.270 (1998) 45-77.
[8] Halil Oruc and George M.Phillips, Explicit factorization of the Vandermonde matrix,Linear Algebra and its Applications.315: 113-123, 2000.
[9] M.Gasca,J.J.Martinez,G.Muhlbach,Computation of rational interpolants with prescribed poles,J.Comput.Appl.Math 26 (1989) 297-309.
[10] P.Junghanns,D.Oestreich,Numerische Losung des Staudammproblems mit Drainage,Z.Angew.Math.Mech. 69 (1989) 83-92.
[11] G.Heinig Inversion of Toeplitz Matrices via Generlized Cauchy Matrices and Rational Interpolations, vol.2,Akademie Verlag,1994, pp. 707-711.
[12] G.Heinig Inversion of generalized Cauchy Matrices and other classes of structured matrices,Linear Algebra for Signal Processing,IMA Vol.in Math.Appl.69 (1994) 95-114.
[13] G.Heinig,A.Bojanczyk,Transformation techniques for Toeplitz and Toeplitz-plus-Hankei structured matrices Ⅰ. transformations, Linear Algebra Appl.254 (1997) 193-226.
[14] J.J.Martfnez and J.M.Pena,Factorizations of Cauchy-Vandermonde matrices, Linear Algebra and its Applications. 284: 229-237, 1998.
[15] Tiio Finck,Georg Heinig and Karla Rost,An inversion Formula and Fast Algorithms for Cauchy-Vandermonde Matrices, Linear Algebra and its Applications. 183: 179-191, 1993.
[16] 王仁宏，朱功勤，有理函数逼近及其应用[M]，北京：科学出版社，2004：85-93．
[17] 王仁宏，梁学章，多元函数逼近[M]，北京：科学出版社，1988：62-69．
[18] 崔蓉蓉，黄有度，一元切触有理插值存在性的判别方法，大学学报，22(4)，2006．
[19] 马锦锦，二元切触有理插值，阜阳师范学院学报(自然科学版)，23(1)，2006．
[20] Z.Vavrin,Confluent Cauchy and Cauchy-Vandermonde Matrices,Linear Algebra and its Applications.258: 271-293, 1997.
[21] 盛中平，广义Vandermonde行列式及其应用，高等学校计算数学学报，1996年9，第3期．
[22] T.T.Ha and J.A.Gibson,A Note on the Determinant of a functional Confluent Vandermonde Matrix and Controllability, Linear Algebra and its Applications. 30: 69-75, 1980.
[23] G. chen and Z. Yang, Bezoutian Representation Via Vandermonde Matrices, Linear Algebra and its Applications. 186: 37-44, 1993.
[24] U.Helmke and P.A.Fuhrman, Bezoutians, Linear Algebra and its Applications. 122-124: 1039-1097, 1989.
[25] G.Sansigre and M.Alvarez,On Bezoutian Reduction with Vandermonde Matrices, Linear Algebra and its Applications. 121: 401-408, 1989.
[26] 杨正宏，胡永建和陈公宁，q-adic范德蒙型矩阵的位移结构与逆公式，北京师范大学学报(自然科学版)，39(2)，2003．
[27] V.Mani and R.Hartwig, Some Properties of the Q-adic Vandermonde Matrix,the Electronic Journal of Linear Algebra.1: 18-33, 1996.