tailieunhanh - numerical mathematics and scientific computation volume 1 Episode 3

Tham khảo tài liệu 'numerical mathematics and scientific computation volume 1 episode 3', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | . Linear Algebra and Matrix Computations 65 are the matrices of the form The matrix JTO A is called a Jordan block. It has one eigenvalue A of multiplicity m to which corresponds only one eigenvector 1 61. The Singular Value Decomposition Let A G RTOXn be a matrix of rank r. Then there is a decomposition of A into a product of three matrices 4 y I 0 I RTOX A Lil. Zj I I G it where u e RTOX and V e Rnxn are orthogonal El diag ơi Ơ2 . ơr and Ơ1 Ơ2 ơr 0. Note that if r n and or r m some of the zero submatrices in E disappear. The ơi are called the singular values of A and if we write u ui um V ui . un the Ui i 1 m and Vi i 1 n are left and right singular vectors respectively. The rank of A equals the number of nonzero singular values. Similarly for any complex matrix A G CTOXn we have the decomposition A IFEVh where u and V are unitary matrices and E a real diagonal matrix. A proof of the singular value decomposition SVD will be given in Sec. . The SVD is of great theoretical and practical The geometrical significance of the SVD can be described as follows. The rectangular matrix A represents a mapping from R to R . From the SVD it follows that there is an orthogonal basis in each of these two spaces with respect to which this mapping is represented by the generalized diagonal matrix E. Note that transposing we obtain the SVD of AT AT V TUT. The singular values of A are uniquely determined. For any distinct singular value ơj Ạ ơi i Ạ j the corresponding singular vector Vj is unique up to a factor 1 . For multiple singular values the corresponding singular vectors can be chosen as any orthonormal basis for the unique subspace that they span. Once the singular 13The SVD was independently published more than a century ago by Eugenio Beltrami 1873 and Camille Jordan 1874. Its use in numerical computations is much more recent. 66 Chapter 1. Principles of Numerical Calculations vectors Vj 1 j r have been chosen the .