申論題

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

(i) (15 %) Letwith r singular valuesU_r= [u₁,⋯,u_r]
consisting of the associated r left singular vectors and V_r= [v₁,⋯,v_r]
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(A^T)) where A^T denotes the transpose of A ；
(2) find the projection matrix P_A such thatand；
(3) find the eignevalues and eigenvectors of AA^T.