MATLAB Function Reference
null

Null space of a matrix

Syntax

• ```Z` `=` `null(A)
Z` `=` `null(A,'r')
```

Description

```Z = null(A) ``` is an orthonormal basis for the null space of `A` obtained from the singular value decomposition. That is, `A*Z` has negligible elements, `size(Z,2)` is the nullity of `A`, and `Z'*Z = I`.

```Z = null(A,'r') ``` is a "rational" basis for the null space obtained from the reduced row echelon form. `A*Z` is zero, `size(Z,2)` is an estimate for the nullity of `A`, and, if `A` is a small matrix with integer elements, the elements of the reduced row echelon form (as computed using `rref`) are ratios of small integers.

The orthonormal basis is preferable numerically, while the rational basis may be preferable pedagogically.

Example

Example 1. Compute the orthonormal basis for the null space of a matrix A.

• ```A = [1     2     3
1     2     3
1     2     3];

Z = null(A)

Z =
0.9636         0
-0.1482   -0.8321
-0.2224    0.5547

A*Z

ans =
1.0e-015 *
0.2220    0.2220
0.2220    0.2220
0.2220    0.2220

Z'*Z

ans =
1.0000   -0.0000
-0.0000    1.0000
```

Example 2. Compute the rational basis for the null space of the same matrix `A`.

• ```ZR = null(A,'r')

ZR =
-2    -3
1     0
0     1

A*ZR

ans =

0     0
0     0
0     0
```

`orth`, `rank`, `rref`, `svd`

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