Find orthonormal basis incrementally 6 27 geometric

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 1 x3 )q1 − (q2 x3 )q2 (remove, q1 , q2 components) q3 = v3 v3 etc. Find orthonormal basis incrementally 6 / 27 Geometric interpretation of Gram-Schmidt process Gram-Schmidt process: v1 = x1 v1 q1 = v1 (normalize) v2 = x2 − (q1 x2 )q1 (remove q1 component from x2 ) q2 = v2 (normalize) v2 v3 = x3 − (q1 x3 )q1 − (q2 x3 )q2 (remove, q1 , q2 components) q3 = v3 v3 etc. ∀i , we have xi = (q1 xi )q1 + (q2 xi )q2 + · · · + (qi −1 xi )qi −1 + vi qi = r1i q1 + r2i q2 + · · · + rii qi 7 / 27 QR decomposition In matrix form, A = QR where A ∈ IRm×n , Q ∈ IRm×n , R ∈ IRn×n : r11 r12 · · · r1n 0 r22 · · · r2n x1 x2 · · · xn = q1 q2 · · · qn . . .. . . . . . . . . 0 A Q 0 ··· rnn R Q Q = I , and R is upper triangular and invertible Usually computed using a variation of Gram-Schmidt process which is less sensitive to numerical errors Can also be computed by Householder transformation or Givens rotations Columns of Q are orthonormal basis in ran(A) Q A=R 8 / 27 General Gram-Schmidt process If x1 , . . . , xn are dependent, we find vj = 0 for some j , which means xj is linearly dependent on x1 , . . . , xj −1 Modified algorithm: when we have vj = 0, skip to the next vector xj +1 and continue k=0 for i = 1, . . . , n { v = xi − k=1 qj qj xi ; j if v = 0 {k = k + 1; qk = } v v }; 9 / 27 Example Consider the matrix 1 1 3 0 2 1 A= 0 0 1 −1 −1 −1 v1 = x1 , q1 = x1 x1 1 1 0 =√ 2 0 −1 0 0 2 1 v2 v2 = x2 − (q1 x2 )q1 = , q2 = = 0 0 v2 0 0 1 1 0 0 v3 1 =√ v3 = x3 − (q1 x3 )q1 − (q2 x3 )q2 = , q3 = 1 v3 3 1 1 1 10 / 27 Example (cont’d) Then we have 1 √ 2 0 R=Q A= 0 1 − √2 0 1 0 0 1 √ 3 0 1 √ 3 1 √ 3 √ √ √ 1 1 3 2 222 0 2 1 − = 0 −2 √1 0 0 1 0 0 3 −1 −1 −1 √ 1 1 1 3 2 0 0 2 1 = A= 0 0 1 0 1 −1 −1 −1 −√ 2 0 −1 0 0 1 √ √ 3 √ √ 2 222 0 0 −2 −1 1 √ √ 3 0 0 3 1 √ 3 11 / 27 Properties of QR decomposition Find orthonormal basis for ran(A) directly Let A = BC with B ∈ IRm×p , C ∈ IRp×n , p = rank(A) To check whether y ∈ span(x1 , . . . , xn ): apply Gram-Schmidt procedure to [x1 , . . ....
View Full Document

This document was uploaded on 02/10/2014.

Ask a homework question - tutors are online