Unexpected type returned from multiplication of adjoint AxisArray

[BSnelling]

Minimal example (using AxisArrays 0.4.3 and julia 1.5):

julia> v = rand(3);

julia> v_ax = AxisArray(v);

julia> typeof(v')
LinearAlgebra.Adjoint{Float64,Array{Float64,1}}

julia> typeof(v_ax')
AxisArray{Float64,2,LinearAlgebra.Adjoint{Float64,Array{Float64,1}},Tuple{Axis{:transpose,Base.OneTo{Int64}},Axis{:row,Base.OneTo{Int64}}}}

The expected return of the following operation would be a scalar.

julia> typeof(v' * rand(3, 3) * v)
Float64

julia> typeof(v_ax' * rand(3, 3) * v_ax)
Array{Float64,1}

But as shown the operation using AxisArray returns an Array. Looking into the methods involved in these operations:

julia> y = rand(3, 3) * v_ax;

julia> @which v_ax' * y
*(A::AbstractArray{T,2}, x::AbstractArray{S,1}) where {T, S} in LinearAlgebra at /Applications/Julia-1.5.app/Contents/Resources/julia/share/julia/stdlib/v1.5/LinearAlgebra/src/matmul.jl:49

The method called by the multiplication of v_ax' and y is:

function (*)(A::AbstractMatrix{T}, x::AbstractVector{S}) where {T,S}
    TS = promote_op(matprod, T, S)
    mul!(similar(x,TS,axes(A,1)),A,x)
end

which shows why the return is an Array. A new method might be needed here so that v_ax' * y returns the expected scalar.

[nickrobinson251]

I think this can be solved by adding a method to AxisArrays to delegate this to *(::Adjoint{Vector}, ::Vector), e.g.

function Base.:*(a::AxisArray{T,2,<:Adjoint{T,<:AbstractVector{T}}}, b::AbstractVector{T}) where {T}
    return *(parent(a), b)
end

we probably need all the same methods as NamedDims has for this
https://github.com/invenia/NamedDims.jl/blob/24b9839091ec4b6091e38c19c9f36e111ca4fbad/src/functions_math.jl#L38-L55