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  (2)

m is a 3×4 matrix:

Use exact arithmetic to find the null space:

Use machine arithmetic:

Use 20-digit precision arithmetic:

Compute the null space for a complex matrix:

Generalizations & Extensions  (1)

Find the null space symbolically:

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:

Introduced in 1988
 (1.0)
 |
Updated in 1996
 (3.0)