Row by column multiplication

Consider two matrices A and B defined by

A=rand(10,3);
B=rand(3,10);

I'm interested in multiplying the vectors defined the first, second, and third rows in B by the vectors defined the first, second, and third columns in A, respectively. My intent is to generate ten, 3X3 matrices defined by

Matrix1= B(:,1)*A(1,:)  
Matrix2= B(:,2)*A(2,:)
...
Matrix10= B(:,10)*A(10,:)

My initial thought was something like

B(:,1:10)*A(1:10,:)

but this approach populates the matrices B and A prior to multiplication, yielding a single 3X3 matrix. How to I change the "order-of-operations" if you will---call each vector on a term-by-term basis, multiplying in between each call to generate the desired matrix? Of course, I really want to avoid having to use for loops.

Answer 1

[Matt J]

Here's a completely vectorized method, which requires James Tursa's MTIMESX function, available at the link below

A=reshape(A',1,3,10);
B=reshape(B,3,1,10);

C=mtimesx(B,A)

http://www.mathworks.com/matlabcentral/fileexchange/25977-mtimesx-fast-matrix-multiply-with-multi-dimensional-support

[John]

Thanks everyone for your prompt comments. I tested each solution by defining

A=rand(10000,3);
B=rand(3,10000);

and nesting each code fragment into 100-iteration for loop. Here were the speeds on my box:

Azzi: 1.46 seconds

Matt Fig's cellfun approach: 7.33 seconds (your prediction was right---the cellfun approach was much slower)

Matt J's cellfun approach: 9.06 seconds

MTIMESX function: 0.156 seconds.

Wow---an order-of-magnitude speed increase over for loops using MTIMESX. That was pretty sick. I think there might be a lot of functionality in this James Tursa code that I need to check out.

Thanks again for all your help!

[Amandeep Gautam]

I wonder if there is anything wrong with A' * B? Seems like it does the same thing. Can you comment?

[Matt J]

That will generate an error

    >> A=rand(10,3); B=rand(3,10); A'*B
    Error using  * 
    Inner matrix dimensions must agree.

Answer 2

[Azzi Abdelmalek]

for k=1:10
  matrix{k}=B(:,k)*A(k,:)
end

%or

for k=1:10
  matrix(:,:,k)=B(:,k)*A(k,:)
end

Answer 3

[Matt Fig]

I think you can avoid FOR loops, but I see no reason to do so here:

A=randi(10,10,3);
B=randi(10,3,10);
for ii = 10:-1:1,C{ii} = B(:,ii)*A(ii,:);end

Here is a vectorization, but I can almost guarantee you it will be slower than the above FOR loop:

cellfun(@mtimes,mat2cell(B,3,ones(1,10)).',mat2cell(A,ones(1,10),3),'Un',0)

Answer 4

[Matt J]

Here's a 1-liner. It's syntactically brief, but not superior to using loops directly,

C=cellfun(@(a,b) a(:)*b(:).',  num2cell(B,1), num2cell(A,2).', 'uni',0)

Answer 5

[Richard Brown]

There is a built in way to do it quickly if you use sparse multiplication. Essentially, you construct the block diagonal matrix blkdiag(B(:, 1), ... , B(:, N)) directly, and multiply it by A to give Y = [B(:, 1)*A(1, :) ; ... ; B(:, N) * A(N, :)]

N = 10000;
A = rand(N, 3);
B = rand(3, N);

I = 1:3*N;
J = repmat(1:N, 3, 1);
Y = mat2cell(sparse(I, J, B) * A, repmat(3, N, 1), 3);

edit: brief explanation added

[Matt J]

Note: both mat2cell and repmat are mfiles that use for-loops, so the solution is not precisely speaking "builtin"

[Richard Brown]

Well, it depends what precisely speaking we mean by "built in" doesn't it?! My definition is that if it comes with a basic MATLAB install it's builtin.

Also, the mat2cell call isn't necessary -- it's just to convert it to the output format in some of the other answers. And the code doesn't touch the branch of repmat that uses for loops. Poor old for loops, they get such a hard time here :)

[Matt J]

I suppose that's true...