R help - Oct 2009 - Why H1=1? (H's the hat matrix)

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!

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!
>
> ______________________________________________
> R-help at 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.
>

Dear Peng,

This seems a curious question to pose on r-help: The vector 1 is the first
column of X, and hence lies in the subspace spanned by the columns of X. H
projects any vector orthogonally onto the subspace spanned by X. Thus, if a
vector, such as 1, lies in this subspace, it's projected onto itself.

Regards,
 John

> -----Original Message-----
> From: r-help-bounces at r-project.org [mailto:r-help-bounces at
r-project.org]
On
> Behalf Of Peng Yu
> Sent: October-11-09 2:03 PM
> To: r-help at stat.math.ethz.ch
> Subject: [R] Why H1=1? (H's the hat matrix)
> 
> 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!
> 
> ______________________________________________
> R-help at 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.