Null Space and Nullity of a matrix in Python

In this tutorial, we would learn about the null space and nullity of a matrix in Python.

Linear relationships among attributes can be found with the help of concepts of Null Space and Nullity.

Null Space in Python

Null Space is the solution obtained from AB = 0 (where A is known matrix and B is a matrix which one needs to find).

First, import the sympy library which is used for symbolic mathematics. Then initialize a list A. Convert it into a matrix using Matrix() and do the same for the null space of A.

Check whether the given condition is satisfied or not by the null space.

from sympy import Matrix 

A = [[1, 2, 3], [4, 5, 6], [7, 8, 9]] 
A = Matrix(A) 
NullSpace = A.nullspace()  
  
NullSpace = Matrix(NullSpace)
print("Null Space : ", NullSpace) 
print(A * NullSpace)

Output-

Null Space : Matrix([[1], [-2], [1]])
Matrix([[0], [0], [0]])

The size of the null space of the matrix provides us with the number of linear relations among attributes.

Nullity of a matrix in Python

Nullity of a matrix A is defined as the size of the null space of the matrix A and so are the linear relations.

First, import the sympy library which is used for symbolic mathematics. Then initialize a list A. Convert it into a matrix using Matrix(). Calculate the number of columns i.e. n and rank of the matrix and then the nullity for the same.

from sympy import Matrix 
A = [[1, 2, 3], [4, 5, 6], [7, 8, 9]] 
A = Matrix(A) 
n = A.shape[1] 
rank = A.rank() 
nullity = n - rank 
print("Nullity : ", nullity)

Output-

Nullity :  1

Rank Nullity Theorem

Nullity of A + Rank(number of linearly independent rows or columns of the matrix) of A = Total number of attributes of A (total number of columns in A)

To see an example of rank, null space, and nullity of a matrix visit-

https://s-mat-pcs.oulu.fi/~mpa/matreng/eem3_2-2.htm

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