Question

IV. Let (10 pts) A= TO0] 4 0 , [ 00] B= 54 0] 3 5. [55] a) Give the reduced SVD decomposition of the matrix A. b) Give the full SVD decomposition of the matrix A. c) Show that 01 = 90 is a singular value of B and find any other singular values if they exist. d) Show that vi = is an eigenvector of B* B corresponding to the singular value 01. e) Give the reduced SVD decomposition of B.

Answer 1

A- --[88] obu ] @ Srp decomposition of A. A = 40 100 AJ!:) Ooo find to Eigenvalues of a 16-10 o 2016- 20 - 2=0.cr x 216 eigenvalues or dao, 16 singlefar value Eigenrector for a2 16 610624. reducer to loool ATA-7) - 1ooo o 160 oo/ eigenveets for 2-16 v for d o Eigenve che ive vz = (3) V-C)

Eigen vector v. (0, 1,0) = length= Vo²+2+0²=1 Normalizing ,-- (0,1,0) for eigen vector 2 (0,0,1)= length='. - Nurmalizing d = (0,0,1) for eigen vector iz 11,0,0) - tength = 1 Mosmalizing az=(1,0,0) Now A'A = 1687 to had eigen volver AA-III = 0 det 2] co-> (16-1) dzo 920, 2216 for x=0 eigenvector - (i) for 2216 eigenvector = lol

for eigenvector 1 Normalizing for eigenvector 2 Nismalizing (0,1 ) length = 1. vi (0) and on (1,0) = length=1. Vico,l). clas 17 TVTG o 140 rooo -OOOO UF4,441 0 0 vi - - AT. u. such that A- U EVT oo U [!] ooo 1 [] 0 0 1 0 1 0 is solution

ruo DI 1 35 Lss BB' - ( 16 12 20 7 2 34 so 2o 4o 5o eigenvalues of BB' = | BB'-d Il=0 11600 12 40 12 341 40 1 20 40 soal = 16) (( 34 ->) (507) J- 40x40) - 12. (12 x (50-1) - 40x 20)+20 (12440 - (34->)820) 20 - --+100? - good zo -(13-100? + good) 20 has solution talo, dago 62 = V10 6 = 90 - 6 = √go is a singular value of B