How to write an explicit one-line code for matrix multiplication A*B, each of size n by n, for example using the summation definition of matrix multiplication (here, eq 14)?
I know how to do it for matrix-vector multiplication: if A is n by n and x is n by 1, then
A*x =sum(A[i,j]*x[j]) for j in 1:n)
A = [1 2; 3 4]
2×2 Matrix{Int64}:
1 2
3 4
x = [1 2 3; 1 2 3]
2×3 Matrix{Int64}:
1 2 3
1 2 3
[sum(A[i, :] .* x[:, j]) for i in 1:size(A, 1), j in 1:size(x, 2)]
2×3 Matrix{Int64}:
3 6 9
7 14 21
Are you looking for something like above !?
where, i lops over each row of A and j loops over each column of B.
Maybe tullio.jl is something worth checking out:
I’ve never used it personally but it seems quite powerful and fast.
I think you just solved a homework problem for the OP.
Just in case someone stumbles onto this answer: the performance can be improved a lot by avoiding copies in the slices and dot product. But of course the best implementation in most cases will be built-in multiplication
Thanks. I have thought of this but it did not help me because I wanted use it as efficient way to incorporate matrices with sparse structures.
What you originally posted sounded very much like a request for solving a homework problem. It would have sounded less like that if you had provided some context/motivation for the request or shown what you had already tried.
Yes, you are right 
I was just solving the problem by keeping in mind the constraints posted in OP (using summation and element-wise multiplication) !!