# Find orthonormal basis incrementally 6 27 geometric

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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 , . . ....
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## This document was uploaded on 02/10/2014.

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