Question

4. Find a QR-factorization of the matrix 5. Find an LU-decomposition of the matrix A =

Answer 1

soln 4) To find QR-factorization of the matrix. we find QR-factorization by using Gram-Schmidt process. A=Q R where Q is orthogonal matrix. - R is upper traingular matrix. Let A=[ail Az Let ui= a1 = (-1,3) ie,=4 lluill - (-1,3) J-14 (352 = t4113) ie e,= (-1 to 3 ) 42 = 92-[92-eige, = (2,-2) – (12,-2).( I )) ( T S ) = (2-2) - Cho tolto io).

= (2,-2)-(是),) - (2,-2)-(-) Uz = (16) log ( 5) | CS)-(台) 三、主(兰,) = (s) 區(台) = . - ( 台) e = ( 6) | avel = (-1.3)-(-13) aven = (2,-2) (完 (十 ) Q1.ex = = )

:.Q = [eid ez] , R= [ave, ale hal which is or matrix. which upper tringular matrix. A- 57 To find LU Decomposition of given : E. = I = [oi] " A=LU we have to find L&W matrix where L is lower tringular matrix. w is Upper tringular matrix. IA=A

107 -27 Loin-1- 21 Rat BRI I JA={: 2] - =) ie. W= (112) Now to find L Let E. =3 ] 15= 1 #0 in Ei is invertible. Eq? become. E, A=U : : A = Eilu : L = Et . E' = 1 Adi Ei - Ten Adiſs i] - thi

.: A = Ei'u ILU Arlon - L3 I Lo