# Unexpected? result of multiplication with adjoint()

**URL:** <https://discourse.julialang.org/t/unexpected-result-of-multiplication-with-adjoint/21447>\
**Category:** Numerics\
**Created:** [March 4, 2019, 11:18am UTC](https://discourse.julialang.org/t/unexpected-result-of-multiplication-with-adjoint/21447 "2019-03-04T11:18:32Z")\
**Posts on this page:** 2\
**Page:** 1

<div class="post-metadata">

**Author:** ![ivanslapnicar](https://avatars.discourse-cdn.com/v4/letter/i/ecb155/32.png) [@ivanslapnicar](https://discourse.julialang.org/u/ivanslapnicar)\
**Post date:** [March 4, 2019, 11:18am UTC](https://discourse.julialang.org/t/unexpected-result-of-multiplication-with-adjoint/21447/1 "2019-03-04T11:18:32Z")

</div>

On `Julia Version 1.0.2` the output of the following code

```julia
using LinearAlgebra
unblock(A) = mapreduce(identity, hcat, [mapreduce(identity, vcat, A[:,i]) for i = 1:size(A,2)])
x=Vector{Any}(undef,2)
x[1]=[1+2im 1-im;-3 4+3im]
x[2]=[1+im 0;1 1+im]
y=unblock(x)
println(x)
println(y)
println(adjoint(y)*y)
println(transpose(conj(x))*x)
println(adjoint(x)==transpose(conj(x)))
println(adjoint(x)*x)

```

is

```julia
Any[Complex{Int64}[1+2im 1-1im; -3+0im 4+3im], Complex{Int64}[1+1im 0+0im; 1+0im 1+1im]]
Complex{Int64}[1+2im 1-1im; -3+0im 4+3im; 1+1im 0+0im; 1+0im 1+1im]
Complex{Int64}[17+0im -12-11im; -12+11im 29+0im]
Complex{Int64}[17+0im -12-11im; -12+11im 29+0im]
true
46 + 0im

```

The correct result is the one obtained by line 9 using the unblocked array `y`.  
The same result is obtained by line 10 using blocked array `x`.  
Although by definition and in Julia adjoint equals conjugate transpose, the result of the last line is scalar.

Multiplication by adjoint works fine is all elements of `x` are scalars. Could it work correctly in the blocked case, too?

---

<div class="post-metadata">

**Author:** ![andreasnoack](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/andreasnoack/32/27_2.png) [@andreasnoack](https://discourse.julialang.org/u/andreasnoack)\
**Post date:** [March 4, 2019, 12:20pm UTC](https://discourse.julialang.org/t/unexpected-result-of-multiplication-with-adjoint/21447/2 "2019-03-04T12:20:25Z")

</div>

See [https://github.com/JuliaLang/julia/pull/27401](https://github.com/JuliaLang/julia/pull/27401) for the change and a discussion of this change. The aim was to make `dot` return a scalar. However, I agree that this is unfortunate would have preferred that the result was a matrix. An example that really makes the issue with the current solution clear is

```julia
julia> x
2-element Array{Any,1}:
 Complex{Int64}[1 + 2im 1 - 1im; -3 + 0im 4 + 3im]
 Complex{Int64}[1 + 1im 0 + 0im; 1 + 0im 1 + 1im]

julia> xm = reshape(x, 2, 1)
2×1 Array{Any,2}:
 Complex{Int64}[1 + 2im 1 - 1im; -3 + 0im 4 + 3im]
 Complex{Int64}[1 + 1im 0 + 0im; 1 + 0im 1 + 1im]

julia> x'x
46 + 0im

julia> (xm'xm)[1]
2×2 Array{Complex{Int64},2}:
  17+0im -12-11im
 -12+11im 29+0im

```

so for array elements, we treat vectors and one-column matrices completely different.

A usecase I’m sad isn’t possible is to store a matrix as a vector of rowvectors.

```julia
julia> X1 = randn(3,2);

julia> X2 = [X1[i,:]' for i in 1:size(X1, 1)];

julia> X3 = reshape(X2, length(X2), 1);

julia> X1'X1
2×2 Array{Float64,2}:
 2.58747 0.733045
 0.733045 0.420716

julia> X2'X2
3.0081843634412744

julia> (X3'X3)[1]
2×2 Array{Float64,2}:
 2.58747 0.733045
 0.733045 0.420716

```

(this would be really useful for `SVector` elememts.)