# Re: [R] about the determinant of a symmetric compound matrix

From: Gabor Grothendieck <ggrothendieck_at_gmail.com>
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
 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]]
>
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