# [R] qr() and Gram-Schmidt

Peter Dalgaard P.Dalgaard at biostat.ku.dk
Mon Nov 3 11:38:46 CET 2008

```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
>
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