set the random seed in julia generator of random numbers

Updated answer, for Julia 0.7 onwards.

import Random
Random.seed!(1234)
dim = 5
A = randn(dim,dim)
H = (A + A')/sqrt(2)

Previous answer, for Julia 0.6 and earlier.

You are looking for the srand function, e.g.

srand(1234)
dim = 5
A = randn(dim,dim)
H = (A + A')/sqrt(2)

Will always produce the same results.