NullSpace

NullSpace[m]

gives a list of vectors that forms a basis for the null space of the matrix m.

Details and Options

  • NullSpace works on both numerical and symbolic matrices.
  • The following options can be given:
  • MethodAutomaticmethod to use
    Modulus 0integer modulus to use
    ToleranceAutomaticnumerical tolerance to use
    ZeroTestAutomaticfunction to test whether matrix elements should be considered to be zero
  • NullSpace[m,Modulus->n] finds null spaces for integer matrices modulo n.
  • NullSpace[m,ZeroTest->test] evaluates test[m[[i,j]]] to determine whether matrix elements are zero.
  • Possible settings for the Method option include "CofactorExpansion", "DivisionFreeRowReduction", and "OneStepRowReduction". The default setting of Automatic switches among these methods depending on the matrix given.

Examples

open allclose all

Basic Examples  (3)

Find the null space of a 3×3 matrix:

The action of m on the vector is the zero vector:

The null space of a symbolic matrix:

Verify that multiplication by m gives the zero vector:

Compute the null space of a rectangular matrix:

Verify that both vectors returned are in the null space:

Scope  (13)

Basic Uses  (8)

Null space of a machine-precision matrix:

Null space of a complex matrix:

Null space of an exact matrix:

Null space of an arbitrary-precision matrix:

Find the null space symbolically:

Null space of non-square matrices:

The null space of a large numerical matrix is computed efficiently:

Null space of a matrix with finite field elements:

Special Matrices  (5)

Null space of a sparse matrix:

Null space of structured matrices:

IdentityMatrix[n] always has an empty null space:

The null space of IdentityMatrix[{m,n}] is nonempty:

Compute the null space for HilbertMatrix:

Compute the null space of a matrix of univariate polynomials of degree :

Options  (1)

Modulus  (1)

m is a 3×3 random matrix of integers between 0 and 4:

Use arithmetic modulo 5 to compute the null space:

The vector is in the null space modulo 5:

Applications  (12)

Spans and Linear Independence  (5)

The following three vectors are not linearly independent:

Therefore the null space of the matrix whose rows are the vectors is nonempty:

The following three vectors are linearly independent:

Therefore the null space of the matrix whose rows are the vectors is empty:

Determine if the following vectors are linearly independent or not:

The matrix formed from the vectors has a nonempty null space, so they are not linearly independent:

Find the dimension of the column space of the following matrix:

Since the null space is empty, the dimension of the column space equals the number of columns:

Find the dimension of the subspace spanned by the following vectors:

Since the matrix rank of the matrix formed by the vectors is three, that is the dimension of the subspace:

Equation Solving and Invertibility  (7)

Determine if the following system of equations has a unique solution:

Rewrite the system in matrix form:

The coefficient matrix has an empty null space, so the system has a unique solution:

Verify the result using Solve:

is a 3×3 singular matrix with a nonempty null space:

Find a solution for :

All solutions are given by , where is any vector in the null space:

Determine if the following matrix has an inverse:

The null space is not trivial, so the matrix is not invertible:

Verify the result using Inverse:

Determine if the following matrix has a nonzero determinant:

Since the null space is empty, its determinant must be nonzero:

Confirm the result using Det:

is an eigenvalue of if the null space of is nontrivial. A matrix is deficient if it has an eigenvalue whose multiplicity is greater than the dimension of the null space of . Show that is an eigenvalue for the following matrix :

Confirm the result using Eigenvalues:

The matrix is deficient because 2 appears twice, but eigenspace is one-dimensional:

Confirm the result with Eigensystem, which indicates deficiency by padding the eigenvector list with zeros:

Find a basis for the eigenvectors of the following matrix:

First, compute the eigenvalues:

For each unique eigenvalue, find the null space:

Combine the two one-dimensional spaces and one two-dimensional space using Join and Apply:

Confirm the result using Eigenvectors:

Estimate the fraction of random 10×10 01 matrix that are invertible:

Properties & Relations  (8)

For any linear combination of elements of the null space of , gives zero:

By the rank-nullity theorem, the dimension of the null space is the number of columns minus MatrixRank[m]:

For a square matrix, m has a trivial null space if and only if Det[m]!=0:

For a square matrix, m has a trivial null space if and only if m has full rank:

For a square matrix, m has a trivial null space if and only if m has an inverse:

For a square matrix, m has a trivial null space iff LinearSolve[m,b] has a solution for a generic b:

The dimension of the null space of a matrix is known as its nullity :

The nullity of a product of two square matrices satisfies Silvester's law of nullity :

In this case, the product in the opposite order has a different nullity:

But it still satisfies the law:

The null space of a square matrix m can be computed using RowReduce:

Do row reduction on the matrix augmented with the identity matrix:

The augmented half of a row is in the null space if the row has a leading 1 in the augmented half:

Get null vectors using NullSpace:

Even though the vectors are not the same, they are a basis for the same vector subspace:

Wolfram Research (1988), NullSpace, Wolfram Language function, https://reference.wolfram.com/language/ref/NullSpace.html (updated 2024).

Text

Wolfram Research (1988), NullSpace, Wolfram Language function, https://reference.wolfram.com/language/ref/NullSpace.html (updated 2024).

CMS

Wolfram Language. 1988. "NullSpace." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2024. https://reference.wolfram.com/language/ref/NullSpace.html.

APA

Wolfram Language. (1988). NullSpace. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/NullSpace.html

BibTeX

@misc{reference.wolfram_2024_nullspace, author="Wolfram Research", title="{NullSpace}", year="2024", howpublished="\url{https://reference.wolfram.com/language/ref/NullSpace.html}", note=[Accessed: 05-December-2024 ]}

BibLaTeX

@online{reference.wolfram_2024_nullspace, organization={Wolfram Research}, title={NullSpace}, year={2024}, url={https://reference.wolfram.com/language/ref/NullSpace.html}, note=[Accessed: 05-December-2024 ]}