Question

Decompentin) Consider A= openConsider A = 32 Denote the 3. Another type of mateix tematice Singeli Vali eigenvalues such that , > tz (a) Calculate AA and AAT (b) Find the eigenvalues and unit eigenvectors of AT A. Respectively denote the eigenvectors by v, and v2. (c) Find the eigenvalues and unit eigenvectors of AAT. Respectively denote the eigenvectors by u, and u. (d) Write down the matrices U = u uz and V = v 02), and S = diag(VI, 12). Calculate UEVT, this should equal to A. This is the singular value decompo- sition of A.

Answer 1

*A=63 AT = 1; } Ma: 1711- 2oC u *** - ;) *** = 1:11:1-6 4 ALAT = 15 Now, 121 14:13 ] Now - M - (50)d +(quo- quo) = 0 d²-5od= 0 dad-su) = o do da su

Now find the eigen veltude. tudo= 0 anda=so. cusode out To H. + 2H2O H = -24 Let Hi 3-24. He it then V illi — eigen- vector Via L.) - wwd-eigen- vectus Cusio a = (autza 10:13 24, + t = 0 He=3# te=at. d, at let then 1:11:}{ -eigen vertone V 1 :) -unit-esign vector

AAT: 15 217 21 45 Now, write it's Charutexeting Eqh!. - (5+45) + (225-941) 5o 12 - Sod + (-210) = d -sud-a16 = (C-54)[0+4) = 0 c = - 4 C2:54 Now. find eigen. Vector fur d. =-4 then! ? ? ? let B= 19 all w r9 217 . Re + 3 R₂ + (-78,1 R₂ ~ +7R-3R, с 10-6 34 + 70,- Hi=- the

w 11 - ww. eigen vector - unit eigen-vector 2 Nettitud dz=54 then = lo ] Rw** 3cm :03 с с - 49 Hit 2112 = 0 - 74, + 3t = 0 3t = 741 H = Isti thon: U= 13 then: U = wel 1unit-eigen-Vectra

نت | ء [ =ه ها eigen-valul find Now, و - ۵ - ( یه +۱) - *) دی (1 ) در ۹ - ۱۰ then' ة = | UE VT = 6:15:11,

С Ел- ЕЛы) ЛЕ - учи ЕЛІ ЕЛ 9-1 ЕЛ - ЕЛ + 1 )