about changing order of Choleski factorization and inverse operation of a matrix

-----Original Message-----
From: r-help-bounces at r-project.org [mailto:r-help-bounces at r-
project.org] On Behalf Of Jie
Sent: Tuesday, July 31, 2012 2:23 PM
To: r-help at r-project.org
Subject: [R] about changing order of Choleski factorization and inverse
operation of a matrix

Dear All,

My question is simple but I need someone to help me out.
Suppose I have a positive definite matrix A.
The funtion chol()  gives matrix L, such that A = L'L.
The inverse of A, say A.inv,  is also positive definite and can be
factorized as A.inv = M'M.
Then
        A = inverse of (A.inv) = inverse of (M'M) = (inverse of M) %*%
(inverse of M)'
           = ((inverse of M)')'%*% (inverse of M)',
i.e. if define B = transpose of (inverse of M), then A = B' %*% B.
Therefore L = B = transpose of (inverse of M) = transpose of (inverse
of
chol(A.inv))
But when I try it in R, the answer is not as expected.

code as below:

 A <- matrix(1:9,3,3)
A <- A + t(A)
diag(A) <- 50
print(A)
L <- chol(A)
B <- t(solve(chol(solve(A))))
print(L)
print(B)

Thank you in advance,

Best wishes,
Jie