Question

10. Find a QR factorization of A 1 A = 4 4 4 -2 2 0

Answer 1

an matrix & R is an uppen friangu To find a QR faclonization of A, where, & is onthogonal matrix. friangular Using Grom - Schmidt process, we have to find out orthonal orthogonal basis of colsen Space (A) = 472 -2 BE > 2 T and let, &, - [1, 1, 1, 1], P 22 = [1, 4,4,-1], &z= [4,-2,2 0] let, ß, = 0, B = dq - C, ß, Where, CIB, is the projection of da upon ß, Then, ß2 is ond Chogonal do ß, and L{B , Be} = {2,4}. (22, B) C, <B., B.) Br = [1, 4, 4, -1]" - 2 [1 1 1 137 [-] 3 2 T whene d, ß, da B2 are projection of Be respectively. Then, Baz is onthogonal to B, B2 and L{ A, A2, A3] = {0, 1, as} B₂2 Lz-d, ß, - d2 B2 Laz upon Bas pesse and 4 d = 23, B) <B,,B) 2 = 1. da = <LB, B2) -O <B2, B2) calling 25 B2 [4,-2,2,0] - [1, 1, 1, 1] + 2 [-08 , & , 3 , - ] [2, -2, 2, -2].

R= 2. 3 2 o 5 -2 O O So, we get the A where, QR factorization of A Ž 2 - 2 2 Q = 2 3 2 - -d Ni and B= 2 2 O 5 -2 2 or . 2 O 0 - Ni- I 2. 2