題組內容

Problem 3. (25 %) Let640158bc43879.jpg denote the set of real m x n matrices and the set of real n x 1 column vectors, respectively, and let640158e3be820.jpg denote the set of real n x n symmetric positive semidefinite (PSD) matrices. This problem includes two parts as follows:

(i) (15 %) Let6401591c29418.jpgwith r singular values6401592c11f8f.jpgUr= [u1,⋯,ur]
consisting of the associated r left singular vectors and Vr= [v1,⋯,vr]
consisting of the associated r right singular vectors.
(1) Find the range space (i.e., column space) of A, denoted as R(A), and the row space of A (i.e., R(AT)) where AT denotes the transpose of A ；
(2) find the projection matrix PA such that64015b2adfebb.jpgand64015b5ae2246.jpg；
(3) find the eignevalues and eigenvectors of AAT.