R help - Sep 2006 - about the determinant of a symmetric compound matrix

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]]

"Stefano Sofia" <stefano.sofia at regione.marche.it> writes:
> 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?
Yes. Unusually, this is not from the famous "Rao p.33", but from
p.32... [1]:

det(A+XX') = det(A)(1+X'A^{-1}X) provided det(A) != 0

now put X = sqrt(c) times a vector of ones and get det(I+cJ) = 1+ck.
Multiply by a^k for the general case. 

Quick sanity check:
> m <- matrix(.1,7,7)
> diag(m) <- .9
> det(m)
[1] 0.393216
> .8^7 * (1 + .1/.8 * 7)
[1] 0.393216

Alternatively, you can do it via eigenvalues: The off-diagonal part
(beta*J) corresponds to a single direction along the unit vector
c(1,1,...,1)/sqrt(7). The diagonal part corresponds to adding (alpha -
beta)*I, which has total sphericity so you can arrange that one
eigenvector of it points in the same direction and you end up with

  (alpha - beta)^(k-1) * (alpha - beta + k*beta) 
> (.9-.1)^6*((.9-.1)+ 7*.1)
[1] 0.393216

(Getting this right on the first try is almost impossible...)

[1] CR Rao, Linear Statistical Inference and Its Applications, 2nd ed.
Wiley 1973.

-- 
   O__  ---- Peter Dalgaard             ?ster Farimagsgade 5, Entr.B
  c/ /'_ --- Dept. of Biostatistics     PO Box 2099, 1014 Cph. K
 (*) \(*) -- University of Copenhagen   Denmark          Ph:  (+45) 35327918
~~~~~~~~~~ - (p.dalgaard at biostat.ku.dk)                  FAX: (+45) 35327907

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]]
>
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