Let QR be such that sqrt(D)X = QR. Then letting solve(...) denote the inverse of ... we have X = solve(sqrt(D))QR which is of the form ZR and Z has the desired weighted orthoginality property.
Since D is diagonal, solve(sqrt(D)) equals diag(1/sqrt(diag(D))) so we get this for Z: diag(1/sqrt(diag(D))) %*% qr.Q(qr(X)) --- Date: Mon, 23 Feb 2004 18:46:34 -0500 From: Stephane DRAY <[EMAIL PROTECTED]> To: <[EMAIL PROTECTED]> Subject: [R] orthonormalization with weights Hello List, I would like to orthonormalize vectors contained in a matrix X taking into account row weights (matrix diagonal D). ie, I want to obtain Z=XA with t(Z)%*%D%*%Z=diag(1) I can do the Gram-Schmidt orthogonalization with subsequent weighted regressions. I know that in the case of uniform weights, qr can do the trick. I wonder if there is a way to do it in the case of non uniform weights by qr or svd ? Thanks in advances. St�phane DRAY -------------------------------------------------------------------------------------------------- D�partement des Sciences Biologiques Universit� de Montr�al, C.P. 6128, succursale centre-ville Montr�al, Qu�bec H3C 3J7, Canada Tel : 514 343 6111 poste 1233 E-mail : [EMAIL PROTECTED] ______________________________________________ [EMAIL PROTECTED] mailing list https://www.stat.math.ethz.ch/mailman/listinfo/r-help PLEASE do read the posting guide! http://www.R-project.org/posting-guide.html
