(*)(::Adjoint{Vector}, ::Vector) should use a non-recursive dot product

[MasonProtter]

This is somewhat tightly related to #22220, but I think a bit different:

julia> σ = [[0  1
             1  0],
            [0 -im
             im 0],
            [1  0
             0  -1]]
3-element Array{Array{Complex{Int64},2},1}:
 [0 + 0im 1 + 0im; 1 + 0im 0 + 0im]
 [0 + 0im 0 - 1im; 0 + 1im 0 + 0im]
 [1 + 0im 0 + 0im; 0 + 0im -1 + 0im]

julia> [1 2 3] * σ
1-element Array{Array{Complex{Int64},2},1}:
 [3 + 0im 1 - 2im; 1 + 2im -3 + 0im]

julia> [1,2,3]'* σ
ERROR: DimensionMismatch("x and y are of different lengths!")
Stacktrace:
 [1] dot at /Users/mason/julia/usr/share/julia/stdlib/v1.4/LinearAlgebra/src/generic.jl:879 [inlined]
 [2] dot(::Array{Int64,1}, ::Array{Array{Complex{Int64},2},1}) at /Users/mason/julia/usr/share/julia/stdlib/v1.4/LinearAlgebra/src/generic.jl:894
 [3] *(::Adjoint{Int64,Array{Int64,1}}, ::Array{Array{Complex{Int64},2},1}) at /Users/mason/julia/usr/share/julia/stdlib/v1.4/LinearAlgebra/src/adjtrans.jl:245
 [4] top-level scope at REPL[46]:1

Here we see that [1 2 3] * σ did something quite different from [1,2,3]'* σ: it essentially did what one should expect given the definitions you see in a linear algebra class, but [1,2,3]'* σ tries to recursively apply dot down to each level of the arrays which produces the above error.

To make matters worse, if I try to 'lift' my vector [1, 2, 3] to the same eltype as σ, I get a disastrously 'wrong' answer:

julia> 𝟙 = [1 0; 0 1];

julia> [1*𝟙, 2*𝟙, 3*𝟙]' * σ
0 + 0im

This seems to come down to the fact that dot is recursive, but I'd argue that even if we want dot to be recursive, * should not be recursive.

Perhaps the solution is a non-recursive dot that * would lower to instead of the recursive form?

This would be unfortunately breaking.

---

Thanks @ericphanson for making clear to me that the source of my issue is the recursive part.

[MasonProtter]

Perhaps even more strange is that transpose actually does the right thing here, but not adjoint, so maybe there is a path to arguing that this change would be a bug fix?

julia> transpose([1,2,3])*σ
2×2 Array{Complex{Int64},2}:
 3+0im   1-2im
 1+2im  -3+0im

(credit to Syx Pek on Slack for noticing this)

[MasonProtter]

#27677