Matrix multiplication : dot product

a= np.array([[2., -1., 0.],[-3.,6.0,1.0]])

array([[ 2., -1.,  0.],
       [-3.,  6.,  1.]])


b= np.array([ [1.0,0.0,-1.0,2],[-4.,3.,1.,0.],[0.,3.,0.,-2.]])

array([[ 1.,  0., -1.,  2.],
       [-4.,  3.,  1.,  0.],
       [ 0.,  3.,  0., -2.]])

np.dot(a,b)

array([[  6.,  -3.,  -3.,   4.],
       [-27.,  21.,   9.,  -8.]])

Dot product of two vectors

Take the first row of above a matrix and the first column of above b matrix:

np.dot( np.array([ 2., -1.,  0.]), np.array([ 1.,-4.,0. ]) )
6.0

Normalize a matrix

Normalize the columns: suppose the columns make up the features, and the rows the observations.

Calculate the 'normalizers':

norms=np.linalg.norm(a,axis=0)

print norms
[ 3.60555128  6.08276253  1. ]

Turn a into normalized matrix an:

an = a/norms

print an

[[ 0.5547002  -0.16439899  0.        ]
 [-0.83205029  0.98639392  1.        ]]