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 Nullity (2.1.0) current version: 3.0.0 »

Source Notebook

Compute the nullity of a matrix

Contributed by: Wolfram|Alpha Math Team

ResourceFunction["Nullity"][mat]

computes the nullity of the matrix mat.

Details and Options

The nullity of a matrix is the dimension of its null space, also called its kernel. The kernel is the space of all input vectors that the matrix maps to zero.

Examples

Basic Examples (3) 

Compute the nullity of a matrix:

In[1]:=
ResourceFunction["Nullity"][{{1, 2}, {3, 4}}]
Out[1]=

Compute the nullity of another matrix:

In[2]:=
ResourceFunction["Nullity"][{{1, 2}, {2, 4}}]
Out[2]=

Compute the nullity of another matrix:

In[3]:=
ResourceFunction["Nullity"][{{0, 0}, {0, 0}}]
Out[3]=

Properties and Relations (5) 

The rank-nullity theorem states that the rank of a matrix plus its nullity equals its column count:

In[4]:=
Grid[Prepend[{MatrixForm[#], MatrixRank[#], ResourceFunction["Nullity"][#], Last[Dimensions[#]]} & /@ {{{1, 2}, {3, 4}}, {{1, 2}, {2, 4}}, {{0, 0}, {0, 0}}, {{1, 2}, {3, 4}, {5, 6}}, {{1, 2, 5}, {2, 4, 6}}, {{1, 2, 3}, {2, 4, 6}, {9, 8, 7}}} , {m, MatrixRank[m], ResourceFunction["Nullity"][m], "number of columns"} ], Frame -> All, Background -> {Automatic, {LightBlue, Automatic}}]
Out[4]=

When called with symbolic arguments, Nullity assumes maximal linear independence:

In[5]:=
ResourceFunction["Nullity"][{{0, 0}, {a, b}}]
Out[5]=

The nullity of an identity matrix is always zero:

In[6]:=
ResourceFunction["Nullity"][IdentityMatrix[5]]
Out[6]=

The nullity of a Hilbert matrix is always zero:

In[7]:=
ResourceFunction["Nullity"][HilbertMatrix[3]]
Out[7]=

The nullity of a Toeplitz matrix is always zero:

In[8]:=
ResourceFunction["Nullity"][ToeplitzMatrix[3]]
Out[8]=

Publisher

Wolfram|Alpha Math Team

Version History

  • 3.0.0 – 23 March 2023
  • 2.1.0 – 12 May 2021
  • 2.0.0 – 24 January 2020
  • 1.0.0 – 17 September 2019

Related Resources

License Information