com.numericalmethod.suanshu.matrix.doubles.linearsystem

Class Kernel

java.lang.Object

com.numericalmethod.suanshu.matrix.doubles.linearsystem.Kernel

All Implemented Interfaces:

java.io.Serializable

public class Kernel
extends java.lang.Object
implements java.io.Serializable

The kernel or null space (also nullspace) of a matrix A is the set of all vectors x for which Ax = 0. The kernel of a matrix with n columns is a linear subspace of n-dimensional Euclidean space. Also, the kernel of a matrix A is exactly the same as the kernel of the linear mapping defined by the matrix-vector multiplication, x → Ax. That is, the set of vectors that map to the zero vector. The rank-nullity theorem says that the rank of A + the dimension of the kernel of A = the number of columns in A.

The kernel is the solution of a system of homogeneous linear equations. With the transformation matrix T, which turns A into the reduced row echelon form, it can solve also a system of non-homogeneous linear equations. Specifically, to find a particular solution for a non-homogeneous system of linear equations with a matrix A, i.e.,

Ax = b

we solve

T * A = U

and then

x = T * b

where x is a particular solution.

See Also:

• Wikipedia: Kernel (matrix)
• Wikipedia: Rank–nullity theorem

, Serialized Form

Nested Class Summary

Nested Classes

Modifier and Type	Class and Description
static class	Kernel.Method the methods available to compute kernel basis.

Constructor Summary

Constructors

Constructor and Description
Kernel(Matrix A) Construct the kernel of a matrix.
Kernel(Matrix A, Kernel.Method method, double epsilon) Construct the kernel of a matrix.

Method Summary

Modifier and Type	Method and Description
java.util.List<Vector>	basis() Get the kernel basis.
java.util.Map<java.lang.Integer,Vector>	basisAndFreeVars() Get the kernel basis and the associated free variables for each basis/column.
boolean	isZero() Check if the kernel has zero dimension, that is, if A has full rank.
int	nullity() Get the nullity of A.
int	rank() Get the rank of A.
Matrix	T() Get the transformation matrix, T, such that T * A = U.
Matrix	U() Get the upper triangular matrix U, such that T * A = U.

Methods inherited from class java.lang.Object

clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

Constructor Detail

Kernel

public Kernel(Matrix A,
              Kernel.Method method,
              double epsilon)

Construct the kernel of a matrix.

Parameters:

A - a matrix

method - the kernel computation method

epsilon - a precision parameter: when a number |x| ≤ ε, it is considered 0; the ε is used to determine the numerical rank of the linear space

Kernel

public Kernel(Matrix A)

Construct the kernel of a matrix.

Parameters:

A - a matrix

Method Detail

nullity

public int nullity()

Get the nullity of A. That is, the dimension of the kernel.

Returns:

the nullity

rank

public int rank()

Get the rank of A. That is, the number of linearly independent columns or rows in A.

Returns:

the rank

basisAndFreeVars

public java.util.Map<java.lang.Integer,Vector> basisAndFreeVars()

Get the kernel basis and the associated free variables for each basis/column. This method is only meaningful when the kernel is computed using Kernel.Method.GAUSSIAN_JORDAN_ELIMINATION. Using this process, a free variable corresponds to an entry '1' in a column. For the example in Wikipedia, x₃ (the 3rd variable) is a free variable, and it corresponds to [3,-5,1,0,0,0]^t.

If the kernel basis is computed using a different method, call basis() instead.

Returns:

a map of indices and kernel basis vectors

See Also:

Wikipedia: Basis

basis

public java.util.List<Vector> basis()

Get the kernel basis.

Returns:

the kernel basis

isZero

public boolean isZero()

Check if the kernel has zero dimension, that is, if A has full rank.

Returns:

true if the kernel is null

T

public Matrix T()

Get the transformation matrix, T, such that T * A = U. To find a particular solution for a non-homogeneous system of linear equations Ax = b, we do

x = T * b

where x is a particular solution.

Returns:

the transformation matrix T

U

public Matrix U()

Get the upper triangular matrix U, such that T * A = U.

Returns:

the U matrix