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
    Modulus0integer 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

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Basic Examples  (1)

Find the null space of a 3×3 matrix:

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

Scope  (10)

Basic Uses  (6)

Null space of a machine-precision matrix:

Null space of a complex matrix:

Null space of an exact matrix:

Find the null space symbolically:

Null space of non-square matrices:

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

Special Matrices  (4)

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 non-empty:

Compute the null space for HilbertMatrix:

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  (2)

m 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:

Find a basis for the eigenspace for a particular eigenvalue:

Properties & Relations  (2)

m is a 5×5 matrix:

The null space of m:

Arbitrary linear combinations of the null space of m give zero:

m is a 3×4 matrix of random zeros and ones:

The MatrixRank equals the column dimension of m minus the dimension of the null space:

Wolfram Research (1988), NullSpace, Wolfram Language function, https://reference.wolfram.com/language/ref/NullSpace.html (updated 1996).

Text

Wolfram Research (1988), NullSpace, Wolfram Language function, https://reference.wolfram.com/language/ref/NullSpace.html (updated 1996).

BibTeX

@misc{reference.wolfram_2021_nullspace, author="Wolfram Research", title="{NullSpace}", year="1996", howpublished="\url{https://reference.wolfram.com/language/ref/NullSpace.html}", note=[Accessed: 05-August-2021 ]}

BibLaTeX

@online{reference.wolfram_2021_nullspace, organization={Wolfram Research}, title={NullSpace}, year={1996}, url={https://reference.wolfram.com/language/ref/NullSpace.html}, note=[Accessed: 05-August-2021 ]}

CMS

Wolfram Language. 1988. "NullSpace." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 1996. 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