[R] about the determinant of a symmetric compound matrix

[Gabor Grothendieck]

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 at 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 at 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.
>