# [R] Why H1=1? (H's the hat matrix)

John Fox jfox at mcmaster.ca
Sun Oct 11 21:19:11 CEST 2009

```Dear Simon,

> -----Original Message-----
> From: r-help-bounces at r-project.org [mailto:r-help-bounces at r-project.org]
On
> Behalf Of Simon Bonner
> Sent: October-11-09 2:33 PM
> To: Peng Yu
> Cc: r-help at stat.math.ethz.ch
> Subject: Re: [R] Why H1=1? (H's the hat matrix)
>
>
> The fitted response for a linear model is a weighted average of the
> observed responses. The i-th row of the hat matrix list the coefficients
> of the average for the i-th fitted value. These values sum to 1 for each
> row, and so H %*% 1=1.

That's true if there's a constant column in X, or more generally if the 1
vector lies in the subspace spanned by the columns of X, but not in general.

Best,
John

>
> Cheers...
>
> -
> Simon Bonner
> Post-Doctoral Fellow
> Department of Statistics, UBC
>
> www.simon.bonners.ca
>
>
> On Sun, 2009-10-11 at 14:09 -0400, Gabor Grothendieck wrote:
> > H projects vectors onto the range of X so any vector already in the
> > range of X gets projected onto itself.
> >
> > On Sun, Oct 11, 2009 at 2:03 PM, Peng Yu <pengyu.ut at gmail.com> wrote:
> > > Suppose I have the following hat matrix:
> > >
> > > H=X(X'X)^{-1}X'
> > > X is a n by p matrix, where n >= p and X_{i,1} = 1
> > >
> > > I'm wondering why H1 = 1. (Here, 1 is column vector, whose each
> > > element is the number 1)
> > >
> > > Thank you!
> > >
> > > ______________________________________________
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