Multiply two matrix by columns with python

Question

I have two matrix:

A = [a11 a12 

     a21 a22]


B = [b11 b12
     b21 b22]

And I want to multiply all its columns (without loops) in order to obtain the matrix:

C =[a11*b11   a11*b12   a12*b11   a12*b12
    a21*b21   a21*b22   a22*b21   a22*b22]

I've tried with

>>> C = np.prod(A,B,axis=0)

but prod doesn't accept two input matrix. Neither np.matrix.prod.

Thanks in advance.

Answer 1

We could use broadcasting for a vectorized solution -

(A[...,None]*B[:,None]).reshape(A.shape[0],-1)

Philosophy : In terms of vectorized/broadcasting language, I would describe this as spreading or putting the second dimension of the input arrays against each other, while keeping their first dimensions aligned. This spreading is done by introducing new axes with None/np.newaxis for these two inputs and then simply multiplying each other.

Mathematical view : Let's use a bit more mathematical view of it with the help of a generic example. Consider input arrays having different number of columns -

In [504]: A = np.random.rand(2,3)

In [505]: B = np.random.rand(2,4)

First off, extend the dimensions and check their shapes -

In [506]: A[...,None].shape
Out[506]: (2, 3, 1)

In [507]: B[:,None].shape
Out[507]: (2, 1, 4)

Now, perform the element-wise multiplication, which will perform these multiplications in a broadcasted manner. Take a closer look at the output's shape -

In [508]: (A[...,None]*B[:,None]).shape
Out[508]: (2, 3, 4)

So, the singleton dimensions (dimension with length = 1) introduced by the use of None/np.newaxis would be the ones along which elements of the respective arrays would be broadcasted under the hood before being multiplied. This under-the-hood broadcasting paired with the respective operation (multiplication in this case) is done in a very efficient manner.

Finally, we reshape this 3D array to 2D keeping the number of rows same as that of the original inputs.

Sample run :

In [494]: A
Out[494]: 
array([[2, 3],
       [4, 5]])

In [495]: B
Out[495]: 
array([[12, 13],
       [14, 15]])

In [496]: (A[...,None]*B[:,None]).reshape(A.shape[0],-1)
Out[496]: 
array([[24, 26, 36, 39],
       [56, 60, 70, 75]])

NumPy matrix type as inputs

For NumPy matrix types as the inputs, we could use np.asmatrix that would simply create view into the inputs. Using those views, the broadcasted element-wise multiplication would be performed, finally resulting in a 2D array after the reshaping. So, the last step would be to convert back to np.matrix type. Let's use the same sample inputs to demonstrate the implementation -

In [553]: A
Out[553]: 
matrix([[2, 3],
        [4, 5]])

In [554]: B
Out[554]: 
matrix([[12, 13],
        [14, 15]])

In [555]: arrA = np.asarray(A)

In [556]: arrB = np.asarray(B)

In [557]: np.asmatrix((arrA[...,None]*arrB[:,None]).reshape(A.shape[0],-1))
Out[557]: 
matrix([[24, 26, 36, 39],
        [56, 60, 70, 75]])