Outer product matrix multiplication - julia

Question

Trying to find the outer product of two matrices A and B
Here is what I have attempted:
function product(A, B)
n_a, m_a = size(A)
n_b, m_b = size(B)
AB = Array{Float64}(undef, n_a, m_b)
for i in 1:n_a
for j in 1:m_b
AB[i, j] = A[:, j] * B[j, :]'
end
end
return AB
end
product(A, B)
I get an error when attempting to run: Cannot `convert` an object of type Matrix{Float64} to an object of type Float64

Answer 1

I'm not exactly sure what you mean by "outer product of matrices", I'm only familiar with an outer product of vectors (which creates a matrix). Could you clarify what output you're looking for with an example?
In any event to address the immediate issue: The stacktrace is pointing to this line in your function:
AB[i, j] = A[:, j] * B[j, :]'
Let's take two example matrices and see what happens for i=j=1:
julia> x = [1 2; 3 4]
2×2 Matrix{Int64}:
1 2
3 4
julia> y = [2 3; 4 5]
2×2 Matrix{Int64}:
2 3
4 5
julia> x[:, 1] * y[1, :]'
2×2 Matrix{Int64}:
2 3
6 9
so the way you are slicing and transposing your matrices means you are calculating an outer product (as I know it) of two vectors, which gives you a matrix. Given that AB is defined as a matrix of Float64s, the location AB[i, j] can only hold a Float64 value, but you are trying to assign a Matrix{Float64} to it.
Again I'm not sure what exactly you are trying to achieve here, but if it's just a "normal" matrix multiplication, you should have
AB[i, j] = A[i, :]' * B[:, j]
in your inner loop. Making that change gives me:
julia> product(x, y) == x * y
true
for the x and y defined above.

Answer 2

Based on the image you linked to in the comment, you're calculating an array of matrices. This can be done as follows:
AB = [A[:,j]*B[j,:]' for j=1:size(A,2)]
Though, I don't believe that's the correct definition for matrix outer product. I think it should be like this:
AB = [r*c' for r in eachrow(A), c in eachcol(B)]
This will give you a [n_a*m_b] matrix of [m_a*n_b] matrices.

Answer 3

From the linked image and the text, I think the formula tries to show another representation for the usual matrix product, which is useful, especially when doing rank decomposition of matrices (as sums of rank one matrices).
As a concrete example, first defining a couple of matrices:
julia> using Random
julia> Random.seed!(1234);
julia> A = rand(1:10,(2,3))
2×3 Matrix{Int64}:
4 3 4
6 9 4
julia> B = rand(1:10,(3,4))
3×4 Matrix{Int64}:
10 8 1 7
8 6 2 10
5 8 5 7
Now define a product as a sum of outer products of vectors, and see it gives the same result as a usual matrix product:
julia> product(A, B) = sum( [ A[:,i] * B[i,:]' for i=1:size(A,2) ] )
product (generic function with 1 method)
julia> product(A,B)
2×4 Matrix{Int64}:
84 82 30 86
152 134 44 160
julia> A*B
2×4 Matrix{Int64}:
84 82 30 86
152 134 44 160
In this example, the resulting matrix can be at most rank 3, the number of columns in A (and rows in B). In fact, it is rank 2 because of number of rows, but in general this product is done on tall times flat matrices and then the rank restriction is meaningful.