# Efficient way to multiply a matrix by each element of another matrix and store in one large matrix

**URL:** <https://discourse.julialang.org/t/efficient-way-to-multiply-a-matrix-by-each-element-of-another-matrix-and-store-in-one-large-matrix/89552>\
**Category:** General Usage\
**Tags:** question\
**Created:** [October 31, 2022, 10:20am UTC](https://discourse.julialang.org/t/efficient-way-to-multiply-a-matrix-by-each-element-of-another-matrix-and-store-in-one-large-matrix/89552 "2022-10-31T10:20:21Z")\
**Posts on this page:** 4\
**Page:** 1

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**Author:** ![CeterisPartybus](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/ceterispartybus/32/46868_2.png) [@CeterisPartybus](https://discourse.julialang.org/u/CeterisPartybus)\
**Post date:** [October 31, 2022, 10:20am UTC](https://discourse.julialang.org/t/efficient-way-to-multiply-a-matrix-by-each-element-of-another-matrix-and-store-in-one-large-matrix/89552/1 "2022-10-31T10:20:21Z")

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I need to multiply a matrix `A` with each element of another matrix `B`. The result should be stored in one large matrix, `C`. This should work for arbitrarily large matrices.

To give a specific example:  
Take to matrices `A`and`B`

```julia
A = [[A11 A12] = [[1 2]
     [A21 A22]] [3 4]]

B = [[B11 B12] = [[11 12]
     [B21 B22]] [13 14]]

```

The result should be:

```julia
C = [[A11*B11 A12*B11 A11*B12 A12*B12] = [[11 22 12 24]
     [A21*B11 A22*B11 A21*B12 A22*B12] [33 44 36 48]
     [A11*B21 A12*B21 A11*B22 A12*B22] [13 26 14 28]
     [A21*B21 A22*B21 A21*B22 A22*B22]] [39 52 42 56]]

```

One solution would be to loop over all elements of `B` but this is probably inefficient and allocating the results of each iteration into matrix `C`might get tricky.

Any better ideas?

I’m sorry if this was asked before. If so, please just refer me to the right topic.

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**Author:** ![Dan](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/dan/32/42581_2.png) [@Dan](https://discourse.julialang.org/u/Dan)\
**Post date:** [October 31, 2022, 11:06am UTC](https://discourse.julialang.org/t/efficient-way-to-multiply-a-matrix-by-each-element-of-another-matrix-and-store-in-one-large-matrix/89552/2 "2022-10-31T11:06:11Z")

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You might want to look at `kron` function.  
As an example:

```julia
julia> A = rand(3:6, 2, 2)
2×2 Matrix{Int64}:
 3 5
 4 6

julia> M = rand(0:2, 3, 3)
3×3 Matrix{Int64}:
 0 0 1
 1 1 2
 2 0 1

julia> kron(M, A)
6×6 Matrix{Int64}:
 0 0 0 0 3 5
 0 0 0 0 4 6
 3 5 3 5 6 10
 4 6 4 6 8 12
 6 10 0 0 3 5
 8 12 0 0 4 6

```

(order of parameters imprtant! second parameter is the ‘small’ matrix tiling the result matrix)

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**Author:** ![stevengj](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/stevengj/32/71_2.png) [@stevengj](https://discourse.julialang.org/u/stevengj)\
**Post date:** [October 31, 2022, 12:09pm UTC](https://discourse.julialang.org/t/efficient-way-to-multiply-a-matrix-by-each-element-of-another-matrix-and-store-in-one-large-matrix/89552/3 "2022-10-31T12:09:47Z")

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What you want is exactly the [Kronecker product](https://en.wikipedia.org/wiki/Kronecker_product) B \otimes A, computed by the [`kron` function](https://docs.julialang.org/en/v1/stdlib/LinearAlgebra/#Base.kron) as noted above.

> [@CeterisPartybus](#):
>
> ```julia
> A = [[A11 A12] = [[1 2]
> [A21 A22]] [3 4]]
> 
> B = [[B11 B12] = [[11 12]
> [B21 B22]] [13 14]]
> 
> ```

These are arrays of arrays, which is not the same thing as a matrix (2d array) in Julia. (Unlike Python.) You want:

```julia
julia> A = [1 2
            3 4]
2×2 Matrix{Int64}:
 1 2
 3 4

julia> B = [11 12
            13 14]
2×2 Matrix{Int64}:
 11 12
 13 14

julia> using LinearAlgebra # for kron

julia> C = kron(B, A)
4×4 Matrix{Int64}:
 11 22 12 24
 33 44 36 48
 13 26 14 28
 39 52 42 56

```

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<div class="post-metadata">

**Author:** ![CeterisPartybus](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/ceterispartybus/32/46868_2.png) [@CeterisPartybus](https://discourse.julialang.org/u/CeterisPartybus)\
**Post date:** [October 31, 2022, 1:30pm UTC](https://discourse.julialang.org/t/efficient-way-to-multiply-a-matrix-by-each-element-of-another-matrix-and-store-in-one-large-matrix/89552/4 "2022-10-31T13:30:49Z")

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Awesome. I knew I should have paid more attention in my linear algebra course.

Thanks a lot!