矩阵值Caratheodory函数广义块Pick矩阵的秩不变性

数学物理学报 ›› 2012, Vol. 32 ›› Issue (1): 103-112.

矩阵值Caratheodory函数广义块Pick矩阵的秩不变性

贺勤¹, 徐清舟², 陈公宁^1,3

1.许昌学院数学与统计学院河南许昌 |461000|
2.许昌学院国际教育学院河南许昌 |461000|
3.北京师范大学数学科学学院北京 100875

收稿日期:2010-08-15 修回日期:2011-04-23 出版日期:2012-02-25 发布日期:2012-02-25
基金资助:
国家自然科学基金(10701009)资助

On Rank Invariance of Generalized Block Pick Matrices of Matrix-valued Caratheodory Functions

HE Qin¹, XU Qing-Zhou², CHEN Gong-Ning^1,3

1.School of Mathematical and Statistics, Xuchang University, Henan Xuchang 461000;
2.School of International Education, Xuchang University, Henan Xuchang 461000;
3.School of Mathematical Sciences, Beijing Normal University, Beijing 100875

Received:2010-08-15 Revised:2011-04-23 Online:2012-02-25 Published:2012-02-25
Supported by:
国家自然科学基金(10701009)资助

摘要/Abstract

摘要：

Lasarow^[1]推导出矩阵值Carath\'{e}odory函数的第一、第二型广义块Pick矩阵及其变型的秩不变性. 这些矩阵由同一个Carath\'{e}odory函数的值与它的直到某阶的导数值确定. 利用文献[2]中提出的块Toeplitz向量方法, 该文断言,这些块矩阵的秩分别相关并重合于具有秩不变性的块Toeplitz矩阵的秩, 从而改进了这两类广义块Pick矩阵的秩不变性结论的证明.

关键词: q ×q Caratheodory函数, 广义块Pick矩阵, 块Toeplitz矩阵, 秩不变性

Abstract:

Lasarow^[1] derived statements on the rank invariance of generalized block Pick matrices of matrix-valued Carath′eodory functions. In this paper, it is concluded that the rank of such generalized block Pick matrices can be related to and coincides with the rank of some block Toeplitz matrices in terms of the Toeplitz vector approach given in [2]. These results are used to improve the proof of the rank invariance of the generalized block Pick matrices in certain sence.

Key words: Matrix-valued Caratheodory function, Generalized block Pick matrix, Block Toeplitz matrix, Rank invariance

中图分类号:

47A57

贺勤, 徐清舟, 陈公宁. 矩阵值Caratheodory函数广义块Pick矩阵的秩不变性[J]. 数学物理学报, 2012, 32(1): 103-112.

HE Qin, XU Qing-Zhou, CHEN Gong-Ning. On Rank Invariance of Generalized Block Pick Matrices of Matrix-valued Caratheodory Functions[J]. Acta mathematica scientia,Series A, 2012, 32(1): 103-112.