Question

Without doing the Gram-Schmidt process again, write a QR decomposition for A' = [2 7 14 2 4 - 7 1 5 4

Answer 1

The given matrix is: A= 2.00 7.00 14.00 2.00 4.00 -7.00 1.00 5.00 4.00 k = 0 ao 2.00 2.00 1.00 -0.67 -0.67 -0.33 Но -0.67 0.73 -0.13 -0.33 -0.13 0.93 Qo -0.67 -0.67 -0.33 -0.67 0.73 -0.13 -0.33 -0.13 0.93 Ao = -3.00 -9.00 -6.00 0.00 -2.40 – 15.00 0.00 1.80 0.00 a = 1.50 B = 0.27 k = 1 -9.00 A1 = -2.40 1.80 1.00 0.00 0.00 Hi = 0.00 -0.80 0.60 0.00 0.60 0.80

-0.67 0.33 -0.67 -0.67 -0.67 0.33 Q1 -0.33 0.67 0.67 -3.00 -9.00 -6.00 A1 0.00 3.00 12.00 0.00 0.00 -9.00 a = 1.25 B = 0.36 k = 2 -6.00 a2 - 12.00 -9.00 1.00 0.00 0.00 H2 0.00 1.00 0.00 0.00 0.00 -1.00 -0.67 0.33 0.67 Q2 -0.67 -0.67 -0.33 -0.33 0.67 -0.67 -3.00 -9.00 -6.00 A2 0.00 3.00 12.00 0.00 0.00 9.00 a = 1.00 ß = 0.50 Result of the QR decomposition

Result of the QR decomposition Q -0.67 0.33 0.67 -0.67 -0.67 -0.33 -0.33 0.67 -0.67 -3.00 -9.00 -6.00 0.00 3.00 12.00 R= 0.00 0.00 9.00 A = Q R = 2.00 7.00 14.00 2.00 4.00 -7.00 1.00 5.00 4.00 QR decomposition with Householder-Method with Rang (A) = n; initialization: A eKmxn AO) = A for k = 1,...,ndo id:= axik (k-1) (k-1) (k-1) V i=k + sign (ax.x 0, d. ak+1, k. .uk) := (0, (k-1) atliekam?) 'm,k K-1 mal · H(K) := Im- (ut na sextasu(k) (w*)); A(K) = H() Ak-1); end Result: A = QR with R := A(n) and Q := H(1)... H(1)... H().
this is QR decomposition for A' by Householder method. Next for A.

The given matrix is: A 1.00 5.00 4.00 2.00 4.00 -7.00 2.00 7.00 14.00 k = 0 1.00 ao = 2.00 2.00 Но -0.33 -0.67 -0.67 -0.67 0.67 -0.33 -0.67 -0.33 0.67 -0.33 -0.67 -0.67 Qo = -0.33 -0.67 0.67 -0.67 -0.33 0.67 -3.00 -9.00 -6.00 0.00 -3.00 - 12.00 Ao = 0.00 0.00 9.00 a = 1.50 ß = 0.33 k= 1 -9.00 a1 = -3.00 0.00 1.00 0.00 0.00 H - 0.00 -1.00 0.00 0.00 0.00 1.00

-0.33 0.67 -0.67 -0.67 -0.67 -0.33 Q1 = -0.67 0.33 0.67 -3.00 -9.00 -6.00 0.00 3.00 12.00 A1 - 0.00 0.00 9.00 a = 1.00 B = 0.50 k= 2 -6.00 a2 12.00 9.00 1.00 0.00 0.00 1.00 0.00 0.00 H2 = 0.00 0.00 – 1.00 -0.33 0.67 0.67 Q2 0.33 -0.67 -0.67 -0.67 0.33 -0.67 A2 -3.00 -9.00 -6.00 0.00 3.00 12.00 0.00 0.00 -9.00 a = 1.00 B = 0.50 Result of the QR decomposition -0.33 0.67 0.67 Q 0.33 -0.67 -0.67 -0.67 0.33 -0.67

Result of the QR decomposition -0.33 0.67 0.67 Q = -0.67 -0.67 0.33 -0.67 0.33 -0.67 -3.00 -9.00 -6.00 R ( 0.00 3.00 12.00 -9.00 0.00 0.00 1.00 5.00 4.00 A = Q R 2.00 4.00 -7.00 2.00 7.00 14.00 QR decomposition with Householder-Method
this is of A.