# [R] orthonormalization with weights

Gabor Grothendieck ggrothendieck at myway.com
Tue Feb 24 04:00:37 CET 2004

```Obviously the R code does not match the
description I gave since we should be taking
the QR decomp of sqrt(D)X, not X.  Z should be:

diag(1/sqrt(diag(D))) %*% qr.Q(qr(sqrt(D)%*%X))

---
Date:   Mon, 23 Feb 2004 21:38:15 -0500 (EST)
From:   Gabor Grothendieck <ggrothendieck at myway.com>
To:   <stephane.dray at umontreal.ca>, <r-help at stat.math.ethz.ch>
Subject:   RE: [R] orthonormalization with weights

diag(1/sqrt(diag(D))) %*% qr.Q(qr(X))

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 <stephane.dray at umontreal.ca>
To: <r-help at stat.math.ethz.ch>
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 ?