X1 x2 xn q1 q2 qn r22 rnn 18

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: angular orthogonalization Steps: v1 = x1 v1 q1 = v1 (normalize) v2 = x2 − (q1 x2 )q1 = (I − q1 q1 )x2 (remove q1 component from x2 ) q2 = v2 (normalize) v2 v3 = x3 − (q1 x3 )q1 − (q2 x3 )q2 = (I − q1 q1 − q2 q2 )x3 = (I − q1 q1 )(I − q2 q2 )x3 (remove, q1 , q2 components) v3 q3 = v3 r11 r12 · · · r1n etc. x1 x2 · · · xn = q1 q2 · · · qn r22 · · · .. . . . . rnn 18 / 27 Gram-Schmidt as triangular orthogonalization (cont’d) x1 x2 · · · xn = q1 q2 · · · qn r11 r12 · · · r22 · · · .. . r1n . . . rnn x1 = r11 q1 x2 = r12 q1 + r22 q2 = r12 x1 + r22 q2 = (q1 x2 )q1 + r22 q2 r11 x3 = r13 q1 + r23 q2 + r33 q3 = (q1 x3 )q1 + (q2 x3 )q2 + r33 q3 . . . xn = r1n q1 + r2n q2 + · · · + rnn qn xi = (q1 xi )q1 + (q2 xi )q2 + · · · + (qi −1 xi )qi −1 + qi qi = r1i q1 + r2i q2 + · · · + rii qi rij = qi xj (i = j ) 19 / 27 Gram-Schmidt as triangular orthogonalization (cont’d) Gram-Schmidt process x1 x2 · · · xn = q1 q2 · · · qn r11 r12 · · · r22 · · · .. . r1n . . . rnn At the j -th step, want to find a unit vector qj ∈ {x1 , . . . , xj } that is orthogonal to q1 , . . . qj −1 vj = xj − (q1 xj )q1 − (q2 xj )q2 − · · · − (qj −1 xj )qj −1 1 q1 = rx11 q2 = . . . x2 −r12 q1 r22 qn = xn − = n −1 i =1 rin qi rnn (I −q1 q1 )x2 (I −q1 q1 )x2 = rij = qi xj (i = j ) |rjj | = xj − j =1 rij qi i = P2 x2 P2 x2 (I −Qn−1 Qn−1 )xn (I −Qn−1 Qn−1 )xn 2 = Pn xn Pn xn (often choose rjj > 0) 20 / 27 Gram-Schmidt as triangular orthogonalization (cont’d) Let A ∈ IRm×n , m ≥ n be a matrix of full rank with columns xi , consider the sequence of formulas q1 = P2 x2 Pn x n P1 x 1 , q2 = , . . . , qn = P1 x1 P2 x2 Pn xn where Pj ∈ IRm×m of rank m − (j − 1) projects x onto the space orthogonal to q1 , . . . , qj −1 (P1 = I when j = 1) Projection Pj can be represented explicitly Let Qj −1 denote the m × (j − 1) matrix containing the first j − 1 columns of Q Qj −1 = q1 q2 · · · qj −1 then Pj = I − Qj −1 Qj −1 21 / 27 Modified Gram-Schmidt process Recall for rank-one orthogonal projection with q ∈...
View Full Document

This document was uploaded on 02/10/2014.

Ask a homework question - tutors are online