Multipling row and column vector using .* operation

Question

a =

1
2
3

b =

1   2   3

a.*b
ans =

1   2   3
2   4   6
3   6   9

I used the .* operator to multiply a row vector and a column vector in Octave to see the results. I dont understand how the answer is obtained.

Answer 1

This is because Octave (in a notable difference from Matlab) automatically broadcasts.

The * operator in Octave is the matrix multiplication operator. So in your case a*b would output (in Matlab as well)

a*b
ans =

1   2   3
2   4   6
3   6   9

which should be expected. The product of a 3-by-1 matrix with a 1-by-3 matrix would have dimensions 3-by-3 (inner dimensions must match, the result takes the outer dimensions).

However the .* operator is the element-wise multiplication operation. That means that instead of matrix multiplication, this would multiply each corresponding elements of the two inputs independent from the rest of the matrix. So [1,2,3].*[1,2,3] (or a'.*b) results in [1,4,9]. Again this is in Matlab and Octave.

When using element-wise operations, it is important that the dimensions of the inputs exactly match. So [1,2,3].*[1,2] will through an error because the dimensions do not match. In Matlab, your a.*b will through an error as well. HOWEVER in Octave it won't, instead it will automatically broadcast. You can imagine this is as if it takes one of your inputs and replicates it on a singleton dimension (so in a column vector, the second dimension is a singleton dimension because it's size is 1) and then applies the operator element-wise. In your case you have two matrices with singleton dimensions (i.e. a columan vector and a row vector) so it actually broadcasts twice and you effectively (but note that it does not actually expand the matrices in memory and is often far faster than using repmat) get

[1,2,3;1,2,3;1,2,3].*[1,1,1;2,2,2;3,3,3]

which produces the result you see.

In matlab, to achieve the same result you would have to explicitly call the bsxfun function (binary singleton expansion function) like so:

bsxfun(@times, a, b)