Linear Algebra - Unit vectors e_i

Specific Domains
15 
struct lazy_e{T} <: AbstractVector{T}
    i::Int
    n::Int
end
lazy_e(i::Int, n::Int, ::Type{T} = Int) where T = lazy_e{T}(i, n)
function Base.getindex(e::lazy_e{T}, i) where T
    @boundscheck @assert i <= e.n
    ifelse(i == e.i, one(T), zero(T))
end
Base.size(e::lazy_e) = (e.n,)

Then on Julia 0.7:

julia> fe(x) = (@inbounds d = x' * lazy_e(3,4); d)
fe (generic function with 1 method)

julia> using StaticArrays, BenchmarkTools

julia> x = @SVector randn(4)
4-element SArray{Tuple{4},Float64,1,4}:
 -0.02571662495110571 
 -0.2620502046882458  
 -0.8456889511406026  
  0.017055806672449558

julia> y = randn(4)
4-element Array{Float64,1}:
 -1.858893446204762 
  1.883085443252701 
 -0.7957485906904918
 -0.5190800039346426

julia> @btime fe($x)
  1.520 ns (0 allocations: 0 bytes)
-0.8456889511406026

julia> @btime fe($y)
  8.301 ns (0 allocations: 0 bytes)
-0.7957485906904918

julia> lazy_e(2,5)
5-element lazy_e{Int64}:
 0
 1
 0
 0
 0

julia> lazy_e(2,5)'
1×5 LinearAlgebra.Adjoint{Int64,lazy_e{Int64}}:
 0  1  0  0  0

EDIT:
Ideally, you’d do this with a for loop and @eval, and perhaps some other helper macros, but:

Base.:*(A::Matrix, e::lazy_e) = A[:, e.i]

Base.:*(A::Adjoint{T,Matrix{T}}, e::lazy_e) where T = A[:, e.i]
Base.:*(A::Transpose{T,Matrix{T}}, e::lazy_e) where T = A[:, e.i]
Base.:*(A::Adjoint{T,Vector{T}}, e::lazy_e) where T = A[e.i]
Base.:*(A::Transpose{T,Vector{T}}, e::lazy_e) where T = A[e.i]


Base.:*(A::Adjoint{T,lazy_e}, e::Matrix{T}) where T = A[e.i, :]
Base.:*(A::Transpose{T,lazy_e}, e::Matrix{T}) where T = A[e.i, :]
Base.:*(A::Adjoint{T,lazy_e}, e::Vector{T}) where T = A[e.i]
Base.:*(A::Transpose{T,lazy_e}, e::Vector{T}) where T = A[e.i]

I had to get really specific for dispatch to actually favor the newly defined multiplication methods, and I’m not sure why. This yielded:

julia> z = randn(3,4)
3×4 Array{Float64,2}:
  0.478915  -0.542889  0.142261  -0.594415
  0.509348   0.63491   1.77353   -0.545189
 -0.659932   0.995295  1.706     -0.291947

julia> @btime $z * lazy_e(2,4)
  26.288 ns (1 allocation: 112 bytes)
3-element Array{Float64,1}:
 -0.5428894820340465
  0.6349099814045577
  0.9952951252882498

julia> @btime $z[:,2]
  26.266 ns (1 allocation: 112 bytes)
3-element Array{Float64,1}:
 -0.5428894820340465
  0.6349099814045577
  0.9952951252882498

The matrix multiplication functions could be edited to return views if you.

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