Connection Between Eigenvectors and Nullspace

Prerequisites:

Some important points about eigenvalues and eigenvectors:

  • Eigenvalues can be complex numbers even for real matrices.
  • When eigenvalues become complex, eigenvectors also become complex.
  • If the matrix is symmetric (e.g A = AT), then the eigenvalues are always real.
  • As a result, eigenvectors of symmetric matrices are also real.
  • There will always be n linearly independent eigenvectors for symmetric matrices.

Now, let’s discuss the connection between eigenvectors and nullspace.

From this article we show that

AX = λX



Now let me ask you a question. What happens when lambda is 0? That is one of the eigenvalues becomes 0.
So, when one of the eigenvalues becomes 0, then we have this equation which is given by

AX = 0 —(equation 1)

From this article we show that

AB = 0 —(equation 2)

So you notice that equation 1 and equation 2 form are the same.

So, that basically means that X which is an eigenvector corresponding to eigenvalue, lambda equals to 0, is a null space vector, because it is just of the form that we have noticed here. So, we could say, the eigenvectors corresponding to zero eigenvalues are in the null space of the original matrix A. Conversely, if the eigenvalue corresponding to an eigenvector is not 0, then that eigenvector can not be in the null space of A. So, these are important results that we need to know.
So, this is how eigenvectors are connected to nullspace.

Example:
Consider the following A matrix

Notice that this is a symmetric matrix hence the eigenvalues are always real as I told before in the important points section.
The eigenvalues for this matrix are

λ = (0, 1, 2)

The eigenvectors corresponding to these eigenvalues are

Code: Python code to calculate eigenvalue and eigenvector

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# Python program to illustrate
# connection between eigenvectors and nullspace
  
# Importing required libraries
import numpy as np
from numpy import linalg
  
# Taking A matrix
A = np.array([
    [0.36, 0.48, 0],
    [0.48, 0.64, 0],
    [0, 0, 2]
])
  
# Calculating eigenvalues and eigenvectors
eigValues, eigVectors = linalg.eig(A)
  
# Printing those values
print("Eigenvalue are :", eigValues)
print("Eigenvectors are :", eigVectors)
  
# Taking eigenvector 1
eigVector1 = np.array([
    [-0.8],
    [0.6],
    [0]
])
  
# Matrix multiplication between A and eigenvector1
result = np.dot(A, eigVector1)
# Print the result
print(result)
  
# This code is contributed by Amiya Rout

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Output:
Eigenvalue are : [0. 1. 2.]
Eigenvectors are : 
[[-0.8 -0.6  0. ]
 [ 0.6 -0.8  0. ]
 [ 0.   0.   1. ]]
[[0.]
 [0.]
 [0.]]

So we have noticed from our discussion before that if X1 is an eigenvector corresponding to lambda equal to 0, then this is going to be in the null space of this matrix A. Let’s verify it by multiplying A with X1. We check that

You can quite easily see that when you do this computation, you will get this (0, 0, 0), which basically shows that this is the eigenvector corresponding to zero eigenvalue.

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