Question

4b

DL for the following 4. Use the LDL Factorization Aleorithm to find a factorization of the form A = LDL for the munces m! 1 T 4 A=1-1 -1 3 - 0 2 b. A = 1 4 0 2 17 03-11 A=2 1 6 3 1 1 1 38 1 4 2 21 2 6 2 1225 1 4 11 1 11 301 1 0 2 1 11-11 4 d. A =

Answer 1

owl a) A- 7 3 0 Li 0.2 This is a Symmetric matrix so we can apply cholesky Decomposition (or) factorization of matrix 14 - I T 14 - I A = 11 30 God! wot Lollo 2 we define A= L XLT where ( aь с Ib dol Io e f loo A = LX LT - n [a² ab ac 7 In 3ol- ab All o 2 lac bitde c²72²762 12 d² betale

I Column 1 a = 4 a=2 Column 2 ab=-| 26=- b = -1 Column ? ac=1 26=1 C=/ ab=-) 6² d 2=3 / betde=0 | Jian - 3 44 Pro 26=-1 b=-1/ d = 1 le= an | betde=o 1²0 76²2 C=1/ +4+ tofa 12 + f²2 72_88-12 ² 44 76 19 f-19

2. O . O dilisha oő 19. L&U 6 1/2 proof multiply L&LT = or 4 i 0 T -3 mail toy Identily matter so we can where o > 10 10 represent 2 o ollo a te 1/27 A= Gooi =

b Sol Given A 4 2 2 6 2 12 2.5 we can respresent this as A = L*LT where y' is lowes & LT is upper triangular matrix , a 007 I la b c b dos L= We have A= L*LT (4 2 27 12 ab ac 6 2 = ab 6 td brede 25 lac betde ctet I Column Clumpa column's a ²-4. I ab=2=26=2 aC=2= 2(=2 a=2 i b= C=1 aC=2 = 1 52+d ² = 6 = 170 = 6 bctde=2= Huse - d = 25 2) e = 2 2 2 2 = 1

ac=2 / betde=2 [c²7e21² 5 Cleate 1+30+=5 f² - 4 - t f² = 19 I f = 19 so 52oo7 12.11 L N5 o L=10 r5 15 [ Vus 5 10 ore so matrix A can be represent in A=LDLT as Zo o o o hin I doo IN5 Ollo illo EU Ti 15 %o o lo

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