In Julia, Transpose operator - julia

Question

I am trying to use a transpose operator over a vector in order to perform am element-wise addition.
For example, I want to add a column vector a = [a1;a2;a3] to a row vector b = [b1,b2] I should get matrix
M = a+b = [a1+b1, a1+b2; a2+b1, a2+b2; a3+b1, a3+b2].
In MATLAB it is equivalent (if both vectors are row vectors) M = a.'+b
I am trying to get the same in Julia but here is the problem, there is no .' operator in Julia starting from 1.0 version. There is the transpose operator which does not work in broadcasting mode. The adjoint operator is not Valid for me because I work with complex numbers.
a = Vector{ComplexF64}([1+3im,2])
b = Vector{ComplexF64}([0,0,0])
Z = zeros(ComplexF64,3,2)
G = zeros(ComplexF64,3,2)
#. Z = b + a' # Works but takes the complex conjugate
#. Z = b + transpose(a) # DOES NOT WORK!!!! The error is " DimensionMismatch("array could not be broadcast to match destination") "
Z = b .+ transpose(a) # Works but not efficient
#. Z = b + conj(a')
The third case Z = b .+ transpose(a) is not efficient because it makes 2 loops first one for addition b .+ transpose(a), than it runs the second loop one for the assignment of b .+ transpose(a) to Z. While the other 3 cases do it within one loop.
So which is the fastest way?
And why transpose doesn't within Broadcasting?
Thank you in advance

Answer 1

For Hermitian You can just type:
a' .+ b
Example
julia> a = ComplexF64.([1+3im,2])
2-element Array{Complex{Float64},1}:
1.0 + 3.0im
2.0 + 0.0im
julia> b = ComplexF64.([10,20,30])
3-element Array{Complex{Float64},1}:
10.0 + 0.0im
20.0 + 0.0im
30.0 + 0.0im
julia> a' .+ b
3×2 Array{Complex{Float64},2}:
11.0-3.0im 12.0+0.0im
21.0-3.0im 22.0+0.0im
31.0-3.0im 32.0+0.0im
If you want to have just transposition you could define your own unary operator (perhaps from the list of unused unary operators):
¬(a) = permutedims(a)
Now you could do
julia> ¬a
1×2 Matrix{ComplexF64}:
1.0+3.0im 2.0+0.0im