The Singular Value Decomposition Trong phần 2, chúng ta đã thấy rằng một ma trận biến đổi vectơ trong phạm vi của nó vào vectơ trong phạm vi của nó (cột không gian), và vector trong không gian vô giá trị của nó vào vector không. Không có vector khác không được ánh xạ vào không gian vô giá trị trái, đó là, vào bổ sung trực giao của phạm vi. Trong phần này, chúng tôi thực hiện tuyên bố này cụ thể hơn bằng cách hiển thị vectors1 đơn vị trong rowspace được biến đổi bởi ma trận | Chapter 3 The Singular Value Decomposition In section 2 we saw that a matrix transforms vectors in its domain into vectors in its range column space and vectors in its null space into the zero vector. No nonzero vector is mapped into the left null space that is into the orthogonal complement of the range. In this section we make this statement more specific by showing how unit vectors1 in the rowspace are transformed by matrices. This describes the action that a matrix has on the magnitudes of vectors as well. To this end we first need to introduce the notion of orthogonal matrices and interpret them geometrically as transformations between systems of orthonormal coordinates. We do this in section . Then in section we use these new concepts to introduce the all-important concept of the Singular Value Decomposition SVD . The chapter concludes with some basic applications and examples. Orthogonal Matrices Consider a point F in R with coordinates Pl p . . . Pn in a Cartesian reference system. For concreteness you may want to think of the case n arguments are general. Given any orthonormal basis Vi . vn for R let but the following 1 q . . _ Qn _ be the vector of coefficients for point F in the new basis. Then for any i . n we have n n vfp vf 52 v 52 VV qi j i j i since the V are orthonormal. This is important and may need emphasis If n p 52 v j i 1 Vectors with unit norm. 23 24 CHAPTER 3. THE SINGULAR VALUE DECOMPOSITION and the vectors of the basis Vi . Vn are orthonormal then the coefficients qj are the signed magnitudes of the projections of p onto the basis vectors Qj yjp. We can write all n instances of equation by collecting the vectors Vj into a matrix V V1 Vn so that q l 7p . Also we can collect the n2 equations T Í if i j V Vi 2 otherwise into the following matrix equation VTV I where is the Il X n identity matrix. Since the inverse of a square matrix 1 is defined as the matrix V 1 such that . I comparison with equation .
