Linear Dependence of Set of Vectors in numpy

Question

I want to check whether some vectors are dependent on each other or not by numpy, I found some good suggestions for checking linear dependency of rows of a matrix in the link below: How to find linearly independent rows from a matrix

I can not understand the 'Cauchy-Schwarz inequality' method which I think is due to lack of my knowledge, however I tried the Eigenvalue method to check linear dependency among columns and here is my code:

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

lambdas, V = np.linalg.eig(A)
print(lambdas)
print(V)

and I get:

[ 1.          0.          1.61803399 -0.61803399]
[[ 0.          0.70710678  0.2763932  -0.7236068 ]
 [ 0.          0.          0.4472136   0.4472136 ]
 [ 0.          0.          0.7236068  -0.2763932 ]
 [ 1.         -0.70710678  0.4472136   0.4472136 ]]

My question is what is the relevance between these eigenvectors or eigenvalues to the dependency of columns of my matrix? How can I understand which columns are dependent to each other and which are independent by these values?

Note that in the linked answer, you should use `np.linalg.eig(A.T)` (the transpose!) to work on the rows — mozway, Jan 24 '22 at 16:22
Your final question doesn't make sense: an eigenvalue does not "belong to" a column of the matrix. Perhaps you want to find a combination of rows that illustrates the linear dependence? Please can you edit your question to clarify what you want. — myrtlecat, Jan 24 '22 at 18:29

score 0 · Accepted Answer · answered Jan 27 '22 at 11:11

The second column vector corresponds to the eigenvalue of 0.

Just take a look at the API documentation when you get confused.

v : (…, M, M) array

The normalized (unit “length”) eigenvectors, such that the column v[:,i] is the eigenvector corresponding to the eigenvalue w[i].

score 0 · Answer 2 · answered Mar 28 '22 at 02:52

0

You can find the linearly independent columns by QR decomposition as described here.

answered Mar 28 '22 at 02:52

Coding elephant

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2

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Linear Dependence of Set of Vectors in numpy

2 Answers2