little_a_and_bigA.R

# Above are some results from the a matrix of a 
# trivariate cholesky with the above diag cells filled in

A = matrix(nrow = 3, byrow = T, c(.71, -.28, .15, -.28, .61, -.38, .15, -.38, .000001))
      [,1]  [,2]     [,3]
[1,]  0.71 -0.28  1.5e-01
[2,] -0.28  0.61 -3.8e-01
[3,]  0.15 -0.38  1.0e-06

# Yeah... so that's wrong. that's the lower a matrix.
# So let's get back to that lower and start again:

a <- A # make a copy
a[upper.tri(a)] <- 0 # empty the upper triangle, and have a look at what we've got
round(a,2)

      [,1]  [,2]  [,3]
[1,]  0.71  0.00    0
[2,] -0.28  0.61    0
[3,]  0.15 -0.38    0

# Now big A is:
A = a %*% t(a) # see how var three now has some variance...
        [,1]    [,2]    [,3]
[1,]  0.5041 -0.1988  0.1065
[2,] -0.1988  0.4505 -0.2738
[3,]  0.1065 -0.2738  0.1669

# And then:
I = diag(3)
rg = solve(sqrt(I * A)) %&% A
round(rg,2)

      [,1]  [,2]  [,3]
[1,]  1.00 -0.42  0.37
[2,] -0.42  1.00 -1.00
[3,]  0.37 -1.00  1.00