null
Null space of matrix
Description
Examples
Null Space of Matrix
Use the null
function to calculate orthonormal and rational basis vectors for the null space of a matrix. The null space of a matrix contains vectors that satisfy .
Create a 3-by-3 matrix of ones. This matrix is rank deficient, with two of the singular values being equal to zero.
A = ones(3)
A = 3×3
1 1 1
1 1 1
1 1 1
Calculate an orthonormal basis for the null space of A
. Confirm that , within roundoff error.
x1 = null(A)
x1 = 3×2
0.8165 -0.0000
-0.4082 -0.7071
-0.4082 0.7071
norm(A*x1)
ans = 4.2999e-16
Now calculate a rational basis for the null space. Confirm that .
x2 = null(A,"rational")
x2 = 3×2
-1 -1
1 0
0 1
norm(A*x2)
ans = 0
x1
and x2
are similar but are normalized differently. While x1'*x1
is an identity matrix, x2'*x2
is not.
x1'*x1
ans = 2×2
1.0000 -0.0000
-0.0000 1.0000
x2'*x2
ans = 2×2
2 1
1 2
Orthogonality is often essential for accuracy of numerical computations. Therefore, the "rational"
option should be used only when working on small all-integer matrices where it is useful for the output to be more readable.
Specify Tolerance for Null Space
When a matrix has small singular values, specify a tolerance to change which singular values are treated as zero.
Create a 7-by-7 Hilbert matrix. This matrix is full rank but has some small singular values.
H = hilb(7)
H = 7×7
1.0000 0.5000 0.3333 0.2500 0.2000 0.1667 0.1429
0.5000 0.3333 0.2500 0.2000 0.1667 0.1429 0.1250
0.3333 0.2500 0.2000 0.1667 0.1429 0.1250 0.1111
0.2500 0.2000 0.1667 0.1429 0.1250 0.1111 0.1000
0.2000 0.1667 0.1429 0.1250 0.1111 0.1000 0.0909
0.1667 0.1429 0.1250 0.1111 0.1000 0.0909 0.0833
0.1429 0.1250 0.1111 0.1000 0.0909 0.0833 0.0769
s = svd(H)
s = 7×1
1.6609
0.2719
0.0213
0.0010
0.0000
0.0000
0.0000
Calculate the null space of H
. Because H
is full rank, Z
is empty.
Z = null(H)
Z = 7x0 empty double matrix
Now, calculate the null space again, but specify a tolerance of 1e-4
. This tolerance leads to null
treating three of the singular values as zeros, so the null space is no longer empty.
Ztol = null(H,1e-4)
Ztol = 7×3
0.0160 -0.0025 0.0002
-0.2279 0.0618 -0.0098
0.6288 -0.3487 0.0952
-0.2004 0.6447 -0.3713
-0.4970 -0.1744 0.6825
-0.1849 -0.5436 -0.5910
0.4808 0.3647 0.1944
Verify that H*Ztol
has negligible elements compared to the specified tolerance.
norm(H*Ztol)
ans = 2.9386e-05
General Solution of Underdetermined System of Equations
Find one particular solution to an underdetermined system, and then obtain the general form for all solutions.
Underdetermined linear systems involve more unknowns than equations. An underdetermined system can have infinitely many solutions or no solution. When the system has infinitely many solutions, they all lie on a line. The points on the line are all obtained with linear combinations of the null space vectors.
Create a 2-by-4 coefficient matrix and use backslash to solve the equation , where is a vector of ones. Backslash calculates a least-squares solution to the problem.
A = [1 8 15 67; 7 14 16 3]
A = 2×4
1 8 15 67
7 14 16 3
b = ones(2,1); x0 = A\b
x0 = 4×1
0
0
0.0623
0.0010
The complete general solution to the underdetermined system has the form , where:
is the null space of .
is any vector of proper length.
is the solution computed by backslash.
Calculate the null space of A
, and then use the result to construct another solution to the system of equations. Check that the new solution satisfies , within roundoff error.
N = null(A)
N = 4×2
-0.2977 -0.8970
-0.6397 0.4397
0.7044 0.0157
-0.0769 -0.0426
x = x0 + N*[1; -2]
x = 4×1
1.4963
-1.5192
0.7354
0.0093
norm(A*x-b)
ans = 2.8908e-14
Input Arguments
A
— Input matrix
matrix
Input matrix.
Data Types: single
| double
Complex Number Support: Yes
tol
— Singular value tolerance
scalar
Singular value tolerance, specified as a real numeric scalar. Singular values of
A
less than the tolerance are treated as zero, which affects the
number of null space vectors returned by null
. The default
tolerance is max(size(A)) * eps(norm(A))
.
Output Arguments
Z
— Null space basis vectors
matrix
Null space basis vectors, returned in the columns of a matrix. Z
satisfies the properties:
A*Z
has negligible elements.size(Z,2)
is an estimate of the nullity ofA
.
If rank(A)
(or rank(A,tol)
) is equal to
size(A,2)
, then Z
is empty.
Algorithms
null(A)
calculates the singular value decomposition of matrix
A
, such that A = U*S*V'
. The columns of
V
corresponding to singular values equal to zero (within tolerance) form
a set of orthonormal basis vectors for the null space.
The rational basis for the null space null(A,"rational")
is obtained
from the reduced row echelon form of A
, as calculated by
rref
.
Extended Capabilities
C/C++ Code Generation
Generate C and C++ code using MATLAB® Coder™.
Usage notes and limitations:
Generated code might return a different basis than MATLAB®.
Code generation does not support the rational basis option (second input).
Code generation does not support sparse matrix inputs for this function.
Thread-Based Environment
Run code in the background using MATLAB® backgroundPool
or accelerate code with Parallel Computing Toolbox™ ThreadPool
.
This function fully supports thread-based environments. For more information, see Run MATLAB Functions in Thread-Based Environment.
GPU Arrays
Accelerate code by running on a graphics processing unit (GPU) using Parallel Computing Toolbox™.
Usage notes and limitations:
The syntax
Z = null(A,"rational")
is not supported.
For more information, see Run MATLAB Functions on a GPU (Parallel Computing Toolbox).
Distributed Arrays
Partition large arrays across the combined memory of your cluster using Parallel Computing Toolbox™.
Usage notes and limitations:
The syntax
Z = null(A,"rational")
is not supported.
For more information, see Run MATLAB Functions with Distributed Arrays (Parallel Computing Toolbox).
Version History
Introduced before R2006aR2022a: Specify tolerance
Use the tol
argument to specify a tolerance threshold for the
singular values used to form the null space. Singular values of the input matrix less than
the tolerance are treated as zero.
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