Numpy: 1D array with various shape

Question

I try to understand how to handle a 1D array (vector in linear algebra) with NumPy.

In the following example, I generate two numpy.array a and b:

>>> import numpy as np
>>> a = np.array([1,2,3])
>>> b = np.array([[1],[2],[3]]).reshape(1,3)
>>> a.shape
(3,)
>>> b.shape
(1, 3)

For me, a and b have the same shape according linear algebra definition: 1 row, 3 columns, but not for NumPy.

Now, the NumPy dot product:

>>> np.dot(a,a)
14
>>> np.dot(b,a)
array([14])
>>> np.dot(b,b)
Traceback (most recent call last):
  File "<stdin>", line 1, in <module>
ValueError: objects are not aligned

I have three different outputs.

What's the difference between dot(a,a) and dot(b,a)? Why dot(b,b) doesn't work?

I also have some differencies with those dot products:

>>> c = np.ones(9).reshape(3,3)
>>> np.dot(a,c)
array([ 6.,  6.,  6.])
>>> np.dot(b,c)
array([[ 6.,  6.,  6.]])

Fred Foo · Accepted Answer · 2013-03-28 15:32:14Z

21

Notice you are not only working with 1D arrays:

In [6]: a.ndim
Out[6]: 1

In [7]: b.ndim
Out[7]: 2

So, b is a 2D array. You also see this in the output of b.shape: (1,3) indicates two dimensions as (3,) is one dimension.

The behaviour of np.dot is different for 1D and 2D arrays (from the docs):

For 2-D arrays it is equivalent to matrix multiplication, and for 1-D arrays to inner product of vectors

That is the reason you get different results, because you are mixing 1D and 2D arrays. Since b is a 2D array, np.dot(b, b) tries a matrix multiplication on two 1x3 matrices, which fails.

With 1D arrays, np.dot does a inner product of the vectors:

In [44]: a = np.array([1,2,3])

In [45]: b = np.array([1,2,3])

In [46]: np.dot(a, b)
Out[46]: 14

In [47]: np.inner(a, b)
Out[47]: 14

With 2D arrays, it is a matrix multiplication (so 1x3 x 3x1 = 1x1, or 3x1 x 1x3 = 3x3):

In [49]: a = a.reshape(1,3)

In [50]: b = b.reshape(3,1)

In [51]: a
Out[51]: array([[1, 2, 3]])

In [52]: b
Out[52]:
array([[1],
       [2],
       [3]])

In [53]: np.dot(a,b)
Out[53]: array([[14]])

In [54]: np.dot(b,a)
Out[54]:
array([[1, 2, 3],
       [2, 4, 6],
       [3, 6, 9]])

In [55]: np.dot(a,a)
---------------------------------------------------------------------------
ValueError                                Traceback (most recent call last)
<ipython-input-55-32e36f9db916> in <module>()
----> 1 np.dot(a,a)

ValueError: objects are not aligned

edited Mar 28, 2013 at 15:32

Fred Foo

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answered Mar 28, 2013 at 11:38

joris

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3 Comments

Marcel Over a year ago

thanks for the answer. Then what is the best practice if I want to do some linear algebra with numpy? Convert all my vector in 2D numpy.array? I'm a bit confused by two different shape...

joris Over a year ago

I am not doing much linear algebra myself, but I think this depends on what you exactly want to do. Just vector products and other simple manipulations/calculations, then 1D is fine. If you want to do matrix calculation etc, you should use 2D.

flonk Over a year ago

The best strategy would be to use 1D arrays for vectors and 2D arrays to represent matrices. Constructions like 2D arrays with shape (1,3) are mainly needed for array magic, but not standard linear algebra stuff. In terms of linear algebra this would correspond to the not very useful object of a 1x3 matrix.

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