cruz wrote: > Hi, > > Why the qr() produces a negative Q compared with Gram-Schmidt? (note > example below, except Q[2,3])
This is a recurrent question in various guises (related to sign issues in factor analysis and PCA). Probably the easiest answer is "Why not?". Notice that at each step of Gram-Schmidt, the sign is really arbitrary: It gives you _an_ orthonormal basis, not the only one. You can't expect a _different_ method for constructing an orthonormal basis to make the _same_ arbitrary choices as G-S! > Here is an example, I calculate the Q by Gram-Schmidt process and > compare the output with qr.Q() > > > a <- c(1,0,1) > b <- c(1,0,0) > c <- c(2,1,0) > x <- matrix(c(a,b,c),3,3) > > ########################## > # Gram-Schmidt > ########################## > > A <- matrix(a,3,1) > q1 <- (1/sqrt(sum(A^2)))*A > B <- b - (q1%*%b)%*%q1 > q2 <- (1/sqrt(sum(B^2)))*B > C <- c - (q1%*%c)%*%q1 - (q2%*%c)%*%q2 > q3 <- (1/sqrt(sum(C^2)))*C > Orthonormal.basis <- matrix(c(q1,q2,q3),3,3) >> Orthonormal.basis > [,1] [,2] [,3] > [1,] 0.7071068 0.7071068 0 > [2,] 0.0000000 0.0000000 1 > [3,] 0.7071068 -0.7071068 0 > > > ########################## > # QR Factorisation X = QR > ########################## > > x.qr <- qr(x) > Q <- qr.Q(x.qr) > R <- qr.R(x.qr) > X <- qr.X(x.qr) >> Q > [,1] [,2] [,3] > [1,] -0.7071068 -0.7071068 0 > [2,] 0.0000000 0.0000000 1 > [3,] -0.7071068 0.7071068 0 > > > Thanks, > cruz > > ______________________________________________ > R-help@r-project.org mailing list > https://stat.ethz.ch/mailman/listinfo/r-help > PLEASE do read the posting guide http://www.R-project.org/posting-guide.html > and provide commented, minimal, self-contained, reproducible code. -- O__ ---- Peter Dalgaard Øster Farimagsgade 5, Entr.B c/ /'_ --- Dept. of Biostatistics PO Box 2099, 1014 Cph. K (*) \(*) -- University of Copenhagen Denmark Ph: (+45) 35327918 ~~~~~~~~~~ - ([EMAIL PROTECTED]) FAX: (+45) 35327907 ______________________________________________ R-help@r-project.org mailing list https://stat.ethz.ch/mailman/listinfo/r-help PLEASE do read the posting guide http://www.R-project.org/posting-guide.html and provide commented, minimal, self-contained, reproducible code.
