Null Space:
The null space of any matrix A consists of all the vectors B such that AB = 0 and B is not zero. It can also be thought as the solution obtained from AB = 0 where A is known matrix of sizem x n
and B is matrix to be found of size n x k
. The size of the null space of the matrix provides us with the number of linear relations among attributes.
A generalized description:
Let a matrix be
1. AB = 0 implies every row of A when multiplied by B goes to zero. 2. Variable values in each sample(represented by a row) behave the same. 3. This helps in identifying the linear relationships in the attributes. 4. Every null space vector corresponds to one linear relationship.
Nullity:
Nullity can be defined as the number of vectors present in the null space of a given matrix. In other words, the dimension of the null space of the matrix A is called the nullity of A. The number of linear relations among the attributes is given by the size of the null space. The null space vectors B can be used to identify these linear relationship. Rank Nullity Theorem: The rank-nullity theorem helps us to relate the nullity of the data matrix to the rank and the number of attributes in the data. The rank-nullity theorem is given by –Nullity of A + Rank of A = Total number of attributes of A (i.e. total number of columns in A)Rank: Rank of a matrix refers to the number of linearly independent rows or columns of the matrix. Example with proof of rank-nullity theorem:
Consider the matrix A with attributes {X1, X2, X3} 1 2 0 A = 2 4 0 3 6 1 then, Number of columns in A = 3This rank and nullity relationship holds true for any matrix. Python Example to find null space of a Matrix:R1 and R3 are linearly independent. The rank of the matrix A which is the number of non-zero rows in its echelon form are 2. we have, AB = 0 Then we get, b1 + 2*b2 = 0 b3 = 0 The null vector we can get is The number of parameter in the general solution is the dimension of the null space (which is 1 in this example). Thus, the sum of the rank and the nullity of A is 2 + 1 which is equal to the number of columns of A.
# Sympy is a library in python for # symbolic Mathematics from sympy import Matrix
# List A A = [[ 1 , 2 , 0 ], [ 2 , 4 , 0 ], [ 3 , 6 , 1 ]]
# Matrix A A = Matrix(A)
# Null Space of A NullSpace = A.nullspace() # Here NullSpace is a list
NullSpace = Matrix(NullSpace) # Here NullSpace is a Matrix
print ( "Null Space : " , NullSpace)
# checking whether NullSpace satisfies the # given condition or not as A * NullSpace = 0 # if NullSpace is null space of A print (A * NullSpace)
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Null Space : Matrix([[-2], [1], [0]]) Matrix([[0], [0], [0]])Python Example to find nullity of a Matrix:
from sympy import Matrix
A = [[ 1 , 2 , 0 ], [ 2 , 4 , 0 ], [ 3 , 6 , 1 ]]
A = Matrix(A)
# Number of Columns NoC = A.shape[ 1 ]
# Rank of A rank = A.rank()
# Nullity of the Matrix nullity = NoC - rank
print ( "Nullity : " , nullity)
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Nullity : 1
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