复矩阵的Givens变换及其QR分解  

Givens transformation and QR factorization of complex matrix

作　　者：杜鹃[1] 冯思臣[1] 

机构地区：[1]成都理工大学管理科学学院,成都610059

出　　处：《成都理工大学学报（自然科学版）》2011年第6期693-696,共4页Journal of Chengdu University of Technology: Science & Technology Edition

基　　金：国家自然科学基金资助项目(10471112);四川省教育厅自然科学重点资助项目(08ZA114)

摘　　要：实矩阵有成熟的三角分解算法,复矩阵尚无好的三角分解算法。为解决复矩阵的三角分解与QR分解问题,采用科学类比,重新拓展定义,演绎计算的方法,给出复Givens矩阵的定义,推导出了复Givens矩阵是酉矩阵,得到了用有限个复Givens变换将一个n维复向量旋转到任何一个给定方向的方法,证明了任何一个非奇异复矩阵能够通过有限次复Givens变换,分解为一个酉矩阵与一个复非奇异上三解矩阵的乘积,利用复Givens变换解决了复矩阵的QR分解问题。There is a mature algorithm of the triangle factorization of the real matrix,but there is not a good one of that of the complex matrix.To solve the problem on the triangle factorization of the complex matrix,This paper uses the scientific analogy,redefinition,extending the definition,and deduction to solve this problem.It gives a definition for the complex Givens matrix.It also deduces that the complex Givens matrix is a Unitary matrix.It gets the algorithm of rotating a complex vector to any fixed direction by finite complex Givens transformations.It proves that any nonsingular complex matrix can be factorized in a product between a Unitary matrix and a complex nonsingular upper triangular matrix by finite complex Givens transformations.It solves the QR factorization of the complex matrix by the complex Givens transformations.

关 键 词：复Givens变换 Givens矩阵 QR分解 

分 类 号：O151.21[理学—数学]