Problem 3. (25 %) Let

denote the set of real m x n matricesand the set of real n x 1 column vectors, respectively, and let

denote the setof real n x n symmetric positive semidefinite (PSD) matrices. This problemincludes two parts as follows:

【題組】

(i) (15 %) Let

with r singular values

Ur= [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), andthe row space of A (i.e., R(AT)) where AT denotes the transpose ofA ；
(2) find the projection matrix PA such that

and

；
(3) find the eignevalues and eigenvectors of AAT.