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# GSQR - Gram-Schmidt and QR The Gram-Schmidt process(GSp...

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Gram-Schmidt and QR The Gram-Schmidt process (GSp) takes a sequence a 1 , a 2 , . . . , a n of linearly independent vectors and gives a sequence q 1 , q 2 , . . . , q n of orthonormal vectors which satisfy Span( q 1 , q 2 , . . . , q k ) = Span( a 1 , a 2 , . . . , a k ) , k = 1 , 2 , . . . , n. (1) Everything about the GSp is here in (1). You can even see how it works: Suppose you have q 1 , q 2 , . . . , q k - 1 , and you want q k . Since we want a k Span( q 1 , q 2 , . . . , q k ), we write a k = r kk q k + k - 1 j =1 r jk q j . (2) Premultiplying by q t j (and noting orthogonality) gives r jk = q t j a k , j = 1 , 2 , . . . , k - 1 , (3) and now that the r ij are known we can use (2) to define w r kk q k = a k - k - 1 j =1 r jk q j . (4) This gives the direction of q k , and r kk is chosen (usually positive) so that q k has unit length: r kk = w 2 , (5) and q k = w/r kk . (6) The GSp is simply (3), (4), (5), and (6) for k = 1 , 2 , . . . , n . The geometry of the k th step of GSp is simple: For each j = 1 , 2 , . . . , k - 1, (3) and (4) subtracts from a k its projection onto q j . The resulting vector is then orthogonal to q j . The final vector, w , is then orthogonal to q 1 , q 2 , . . . , q k - 1 . Interpret r jk
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