Sum under main diagonal in julia

You're looking for the LinearAlgebra module, which is part of the standard library and contains a tril function:

julia> using LinearAlgebra

julia> A = ones(5, 5)
5×5 Array{Float64,2}:
 1.0  1.0  1.0  1.0  1.0
 1.0  1.0  1.0  1.0  1.0
 1.0  1.0  1.0  1.0  1.0
 1.0  1.0  1.0  1.0  1.0
 1.0  1.0  1.0  1.0  1.0

julia> tril(A, -1)
5×5 Array{Float64,2}:
 0.0  0.0  0.0  0.0  0.0
 1.0  0.0  0.0  0.0  0.0
 1.0  1.0  0.0  0.0  0.0
 1.0  1.0  1.0  0.0  0.0
 1.0  1.0  1.0  1.0  0.0

julia> sum(tril(A, -1))
10.0

tril function is in the Linear Algebra standard library. To use it, you need to import the LinearAlgebra module. tril takes a second optional argument to specify where you want to start the lower triangular part. You can set it to -1 in order not to take the main diagonal.

using LinearAlgebra

A = rand(5, 5)
lowerA = tril(A, -1)
println(sum(lowerA)) 

You might also want to take a look at lazy lower triangular view constructors such as LowerTriangular or OneUnitLowerTriangular.