Broadcasting in matrix multiplication

Consider a 1x2 matrix A and a 1d array of 2x2 matrices B:

A = [1 2]
B = [rand(2,2) for _ in 1:3]

Obviously, B[1] is a 2x2 matrix, B[2] and B[3] as well.

Now, I would like to get an array of matrix products [A*B[1], A*B[2], ... A*B[end]]. How can I get it? I am trying the dot in

julia> R = A.*B
3×2 Array{Array{Float64,2},2}:
 [0.460514 0.415326; 0.0842543 0.33096]  [0.921028 0.830652; 0.168509 0.66192]
 [0.908421 0.342417; 0.103733 0.155627]  [1.81684 0.684833; 0.207466 0.311255]
 [0.828274 0.451481; 0.053429 0.810234]  [1.65655 0.902963; 0.106858 1.62047]

but obviously it is not the expected result

julia> R[1]
2×2 Array{Float64,2}:
 0.460514   0.415326
 0.0842543  0.33096

Instead, R[1] (and for other indices as well) is expected to be a 1x2 matrix.

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You want Ref(A).*B. Ref is a single element container, so broadcast will broadcast over it, not A.


Wow, that was superfast :slight_smile: Thanks. Now that I have a solution, I still have to understand it. This Ref stuff is completely new to me. Thanks.

It’s conceptually the same as [A].*B You are basically just creating a very low overhead container, so that A is treated as an element rather than a collection.


You can also just use (A,) (a one element tuple). I personally like it better than Ref because I thought that was mostly meant for ccall usage.


You could also use a comprehension, which is clearer, IMO, or even a generator if you don’t want to materialize.

julia> R = [A*b for b in B]
3-element Array{Array{Float64,2},1}:
 [1.1064518582166465 1.4257048365761784]
 [0.7462570773129014 0.2360058367114488]
 [0.2639640481443961 1.0567511644538756]

julia> R = (A*b for b in B)
Base.Generator{Array{Array{Float64,2},1},var"#11#12"}(var"#11#12"(), [[0.28439915229561796 0.024986306638530964; 0.41102635296051426 0.7003592649688237], [0.6502316144726392 0.15084252545701005; 0.04801273142013107 0.042581655627219384], [0.0710881212956167 0.14551768207448657; 0.0964379634243897 0.45561674118969453]])

Excellent, thanks. Indeed, it is clearer.

It is indeed clearer. In the case A and each element in B are square matrices, how could you do it inplace?

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B .= [A*b for b in B] works, but does do some allocations

for 2x2 this seems to work

using LinearAlgebra
for b in B

but when it gets bigger it attempts calling BLAS which doesn’t like the output being an alias to the input

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I suspect that this may actually be worse than B = [A*b for b in B], because either way it allocates on the right-hand side, but then it additionally copies over to the left hand side instead of just rebinding the variable, which is basically free.

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True, but it probably depends on the relative sizes of B and A and memory pressure etc. One big allocation for a new B, might or might not be relevant compared to the numerous “small” allocations for A.
Also the inplace version might still be relevant if for example B is a view into a larger array. But now I’m probably reaching :slight_smile:
Either way, the request was for an inplace version and I was hoping someone would suggest a clever way to fix the allocations. (The fact that the for loop with the mul! gave different behaviour for different sizes also thought me something new today).

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Actually, my (untested) thesis is that the dotted assignment is strictly worse than the undotted one, because the dotted one does everything that the undotted one does, plus extra copying at the end. I think.

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