# X1 x2 xn q1 q2 qn r22 rnn 18

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

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