"Stefano Sofia" <[EMAIL PROTECTED]> 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 Ă˜ster Farimagsgade 5, Entr.B c/ /'_ --- Dept. of Biostatistics PO Box 2099, 1014 Cph. K (*) \(*) -- University of Copenhagen Denmark Ph: (+45) 35327918 ~~~~~~~~~~ - ([EMAIL PROTECTED]) FAX: (+45) 35327907 ______________________________________________ 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.