Vectorspace Orthonormalization

A function that converts a set of linearly independent vectors to a set of orthonormal basis vectors.

from mlxtend.math import vectorspace_orthonormalization

Overview

The vectorspace_orthonormalization converts a set linearly independent vectors to a set of orthonormal basis vectors using the Gram-Schmidt process [1].

References

  • [1] https://en.wikipedia.org/wiki/Gram–Schmidt_process

Example 1 - Convert a set of vector to an orthonormal basis

Note that to convert a set of linearly independent vectors into a set of orthonormal basis vectors, the vectorspace_orthonormalization function expects the vectors to be arranged as columns of a matrix (here: NumPy array). Please keep in mind that the vectorspace_orthonormalization function also works for non-linearly independent vector sets; however, the resulting vectorset won't be orthonormal as a result. An easy way to check whether all vectors in the input set are linearly independent is to use the numpy.linalg.det (determinant) function.

import numpy as np
from mlxtend.math import vectorspace_orthonormalization

a = np.array([[2,   0,   4,  12],
              [0,   2,  16,   4],
              [4,  16,   6,   2],
              [2, -12,   4,   6]])


s = ''
if np.linalg.det(a) == 0.0:
    s = ' not'
print('Input vectors are%s linearly independent' % s)


vectorspace_orthonormalization(a)
Input vectors are linearly independent





array([[ 0.40824829, -0.1814885 ,  0.04982278,  0.89325973],
       [ 0.        ,  0.1088931 ,  0.99349591, -0.03328918],
       [ 0.81649658,  0.50816781, -0.06462163, -0.26631346],
       [ 0.40824829, -0.83484711,  0.07942048, -0.36063281]])

Note that scaling the inputs equally by a factor should leave the results unchanged:

vectorspace_orthonormalization(a/2)
array([[ 0.40824829, -0.1814885 ,  0.04982278,  0.89325973],
       [ 0.        ,  0.1088931 ,  0.99349591, -0.03328918],
       [ 0.81649658,  0.50816781, -0.06462163, -0.26631346],
       [ 0.40824829, -0.83484711,  0.07942048, -0.36063281]])

However, in case of linear dependence (the second column is a linear combination of the first column in the example below), the vector elements of one of the dependent vectors will become zero. (For a pair of linear dependent vectors, the one with the larger column index will be the one that's zero-ed.)

a[:, 1] = a[:, 0] * 2
vectorspace_orthonormalization(a)
array([[ 0.40824829,  0.        ,  0.04155858,  0.82364839],
       [ 0.        ,  0.        ,  0.99740596, -0.06501108],
       [ 0.81649658,  0.        , -0.04155858, -0.52008861],
       [ 0.40824829,  0.        ,  0.04155858,  0.21652883]])

API

vectorspace_orthonormalization(ary, eps=1e-13)

Transforms a set of column vectors to a orthonormal basis.

Given a set of linearly independent vectors, this functions converts such column vectors, arranged in a matrix, into orthonormal basis vectors.

Parameters

  • ary : array-like, shape=[num_vectors, num_vectors]

    A set of vectors (arranged as columns in a matrix)

  • eps : float (default: 1e-13)

    A small tolerance value to determine whether the vector norm is zero or not.

Returns

  • arr : array-like, shape=[num_vectors, num_vectors]

    An orthonormal set of vectors (arranged as columns)