Simplest solution

Use numpy.dot or a.dot(b). See the documentation here.

Use with AI>>> a = np.array([[ 5, 1 ,3], 
                  [ 1, 1 ,1], 
                  [ 1, 2 ,1]])
>>> b = np.array([1, 2, 3])
>>> print a.dot(b)
array([16, 6, 8])

This occurs because numpy arrays are not matrices, and the standard operations *, +, -, / work element-wise on arrays.

Note that while you can use numpy.matrix (as of early 2021) where * will be treated like standard matrix multiplication, numpy.matrix is deprecated and may be removed in future releases.. See the note in its documentation (reproduced below):

It is no longer recommended to use this class, even for linear algebra. Instead use regular arrays. The class may be removed in the future.

Thanks @HopeKing.

Other Solutions

Also know there are other options:

• As noted below, if using python3.5+ and numpy v1.10+, the @ operator works as you'd expect:

  Use with AI>>> print(a @ b)
  array([16, 6, 8])
  

• If you want overkill, you can use numpy.einsum. The documentation will give you a flavor for how it works, but honestly, I didn't fully understand how to use it until reading this answer and just playing around with it on my own.

  Use with AI>>> np.einsum('ji,i->j', a, b)
  array([16, 6, 8])
  

• As of mid 2016 (numpy 1.10.1), you can try the experimental numpy.matmul, which works like numpy.dot with two major exceptions: no scalar multiplication but it works with stacks of matrices.

  Use with AI>>> np.matmul(a, b)
  array([16, 6, 8])
  

• numpy.inner functions the same way as numpy.dot for matrix-vector multiplication but behaves differently for matrix-matrix and tensor multiplication (see Wikipedia regarding the differences between the inner product and dot product in general or see this SO answer regarding numpy's implementations).

  Use with AI>>> np.inner(a, b)
  array([16, 6, 8])
  
  # Beware using for matrix-matrix multiplication though!
  >>> b = a.T
  >>> np.dot(a, b)
  array([[35,  9, 10],
         [ 9,  3,  4],
         [10,  4,  6]])
  >>> np.inner(a, b) 
  array([[29, 12, 19],
         [ 7,  4,  5],
         [ 8,  5,  6]])
  

• If you have multiple 2D arrays to dot together, you may consider the np.linalg.multi_dot function, which simplifies the syntax of many nested np.dots. Note that this only works with 2D arrays (i.e. not for matrix-vector multiplication).

  Use with AI  >>> np.dot(np.dot(a, a.T), a).dot(a.T)
    array([[1406,  382,  446],
           [ 382,  106,  126],
           [ 446,  126,  152]])
    >>> np.linalg.multi_dot((a, a.T, a, a.T))
    array([[1406,  382,  446],
           [ 382,  106,  126],
           [ 446,  126,  152]])
  

Rarer options for edge cases

• If you have tensors (arrays of dimension greater than or equal to one), you can use numpy.tensordot with the optional argument axes=1:

  Use with AI>>> np.tensordot(a, b, axes=1)
  array([16,  6,  8])
  

• Don't use numpy.vdot if you have a matrix of complex numbers, as the matrix will be flattened to a 1D array, then it will try to find the complex conjugate dot product between your flattened matrix and vector (which will fail due to a size mismatch n*m vs n).