From: Gabor Grothendieck <ggrothendieck_at_gmail.com>

Date: Wed 27 Sep 2006 - 06:14:46 GMT

R-help@stat.math.ethz.ch 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. Received on Wed Sep 27 16:20:22 2006

Date: Wed 27 Sep 2006 - 06:14:46 GMT

If P = projection onto the one dimensional space spanned by 1, the vector consisting of n 1's, then using the usual formula for projections we have P = 11'/1'1 = J/n

and writing I+cJ in terms of P we have:

I+cJ = (I-P) + (cn+1)P

which is an eigen expansion showing that
I+cJ has one eigenvalue of cn+1 and n-1

eigenvalues of 1 so its determinant, being
the product of the eigenvalues, is cn+1.
That is,

det(I+cJ) = cn+1

and we can verify that for n=5 and c=10

which should give cn+1 = 51:

*> det(diag(5) + matrix(10, 5, 5)) # 10 * 5 + 1 = 51
*

[1] 51

Thus det(a(I+cJ)) = a^n (cn+1)

On 9/26/06, Stefano Sofia <stefano.sofia@regione.marche.it> wrote:

> Dear R users,

*> even if this question is not related to an issue about R, probably some of you will be able to help me.
**>
**> I have a square matrix of dimension k by k with alpha on the diagonal and beta everywhee else.
**> This symmetric matrix is called symmetric compound matrix and has the form
**> a( I + cJ),
**> where
**> I is the k by k identity matrix
**> J is the k by k matrix of all ones
**> a = alpha - beta
**> c = beta/a
**>
**> I need to evaluate the determinant of this matrix. Is there any algebric formula for that?
**>
**> thank you for your help
**> Stefano
**>
**>
**>
**> [[alternative HTML version deleted]]
**>
**> ______________________________________________
**> R-help@stat.math.ethz.ch 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.
**>
*

R-help@stat.math.ethz.ch 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. Received on Wed Sep 27 16:20:22 2006

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