DiegoDZ DiegoDZ - 17 days ago 9
Python Question

Multiply two matrix by columns with python

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

You 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 dimension 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 less a bit more mathematical view of it with the help of a generic example that has different number of columns -

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

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

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.

Finally, we reshape this 3D array to have a 2D array with 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]])

For NumPy matrix type inputs

If you are working NumPy matrix types, you 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. Thus, for np.matrix input, the solution would be -

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]])