[FIXED] Effect of reshaping an numpy array from (2,) to (2,1)

Issue

I was trying to calculate a covariance matrix of 2 arrays a and b via a dot product in python with the formula a @ b.T.

The initial form of my input arrays are (2,) when the shape is printed. Obviously the result of this calculation is a single scalar but not the covariance matrix i need.
As it turns out i needed to reshape the arrays(i.e. a.reshape(2,1)) to a receive the desired (2,2) covariance matrix.

Now my problem is that i dont understand what effect the reshaping of the arrays has on the data, more specifically why i am allowed to do it and receive the result i want.
How come i can just reshape the array to receive the covariance matrix, what is the thought process behind it?
I do understand the how the dot product works, but as i said i dont get why i am allowed to just reshape the array to receive the desired outcome dimensions.

I hope i made my problem understandable, i have a bit of trouble formulating it.

Appreciate any help!

Solution

While reshape doesn't change the values of an array, it does affect how the array interacts with other arrays. Operators like * and @ have their rules about handling arrays with various shapes.

In [221]: a, b = np.array([1,2]), np.array([3,4])
In [222]: a,b
Out[222]: (array([1, 2]), array([3, 4]))

transpose of a 1d array changes nothing. It does not 'add' a dimension; it just reverses the existing dimensions:

In [223]: b.T
Out[223]: array([3, 4])

For 1d arrays, matmul and dot do the 'scalar', or 'dot' product. That's clearly documented:

In [224]: a@(b.T)
Out[224]: 11

Reshaping the arrays to (2,1). Now transpose does make a difference:

In [225]: a[:,None]
Out[225]: 
array([[1],
       [2]])
In [226]: b[:,None].T
Out[226]: array([[3, 4]])

Here we have a (2,1) with (1,2), which does the matrix product, summing on the size 1 dimensions. The result is (2,2). Again clearly documented.

In [227]: a[:,None]@b[:,None].T
Out[227]: 
array([[3, 4],
       [6, 8]])

Other ways of getting that product:

In [228]: np.outer(a,b)
Out[228]: 
array([[3, 4],
       [6, 8]])
In [229]: a[:,None]*b
Out[229]: 
array([[3, 4],
       [6, 8]])

The last case does elementwise multiplication of (2,1) with a (2,). The rules of broadcasting apply. Read up on that!

Answered By - hpaulj