How to find linearly independent rows from a matrix

Question

How to identify the linearly independent rows from a matrix? For instance,

The 4th rows is independent.

Answer 1

First, your 3rd row is linearly dependent with 1t and 2nd row. However, your 1st and 4th column are linearly dependent.

Two methods you could use:

Eigenvalue

If one eigenvalue of the matrix is zero, its corresponding eigenvector is linearly dependent. The documentation eig states the returned eigenvalues are repeated according to their multiplicity and not necessarily ordered. However, assuming the eigenvalues correspond to your row vectors, one method would be:

import numpy as np

matrix = np.array(
    [
        [0, 1 ,0 ,0],
        [0, 0, 1, 0],
        [0, 1, 1, 0],
        [1, 0, 0, 1]
    ])

lambdas, V =  np.linalg.eig(matrix.T)
# The linearly dependent row vectors 
print matrix[lambdas == 0,:]

Cauchy-Schwarz inequality

To test linear dependence of vectors and figure out which ones, you could use the Cauchy-Schwarz inequality. Basically, if the inner product of the vectors is equal to the product of the norm of the vectors, the vectors are linearly dependent. Here is an example for the columns:

import numpy as np

matrix = np.array(
    [
        [0, 1 ,0 ,0],
        [0, 0, 1, 0],
        [0, 1, 1, 0],
        [1, 0, 0, 1]
    ])

print np.linalg.det(matrix)

for i in range(matrix.shape[0]):
    for j in range(matrix.shape[0]):
        if i != j:
            inner_product = np.inner(
                matrix[:,i],
                matrix[:,j]
            )
            norm_i = np.linalg.norm(matrix[:,i])
            norm_j = np.linalg.norm(matrix[:,j])

            print 'I: ', matrix[:,i]
            print 'J: ', matrix[:,j]
            print 'Prod: ', inner_product
            print 'Norm i: ', norm_i
            print 'Norm j: ', norm_j
            if np.abs(inner_product - norm_j * norm_i) < 1E-5:
                print 'Dependent'
            else:
                print 'Independent'

To test the rows is a similar approach.

Then you could extend this to test all combinations of vectors, but I imagine this solution scale badly with size.