Question

I have two n-by-m matrices, A and B. I want to create a new matrix C which is something like:

for i = 1:n
    C = C + outerProduct(A(i,:), B(i,:));
end

i.e. C is a matrix of size m x m, the sum of all outer products of the rows of A and B.

Is there a fast way to do it without a for loop (given that for loops are notoriously slow in Matlab)?

Answer 1

The operation you are performing (the sum of the row outer products) is equivalent to the multiplication of a transposed version of A with B:

C = A.'*B;

You can see this using the following example:

>> mat = magic(5);  %# A sample 5-by-5 matrix
>> A = mat(1:4,:);  %# Create a 4-by-5 matrix
>> B = mat(2:5,:);  %# Create another 4-by-5 matrix

>> C = zeros(5);  %# Initialize C to be 5-by-5
>> for i = 1:4, C = C + A(i,:).'*B(i,:); end;  %'# Calculate C as you are now

>> isequal(C, A.'*B)  %'# Test for equality with the shorter solution

ans =

     1  %# Equal!