Matrix.GetLeftNullSpace Method

Definition

Namespace: Numerics.NET
Assembly: Numerics.NET (in Numerics.NET.dll) Version: 9.1.2

Overload List

GetLeftNullSpace<T>(Matrix<T>) Computes an orthogonal basis for the left null space of a matrix.
GetLeftNullSpace<T>(Matrix<T>, T) Computes an orthogonal basis for the left null space of a matrix.

GetLeftNullSpace<T>(Matrix<T>)

Computes an orthogonal basis for the left null space of a matrix.
C# 
public static Matrix<T> GetLeftNullSpace<T>(
	this Matrix<T> matrix
)

Parameters

matrix  Matrix<T>
The matrix whose left null space is to be computed.

Type Parameters

T
The type of the elements of the matrix.

Return Value

Matrix<T>
A matrix representing the left null space of the input matrix.

Usage Note

In Visual Basic and C#, you can call this method as an instance method on any object of type Matrix<T>. When you use instance method syntax to call this method, omit the first parameter. For more information, see Extension Methods (Visual Basic) or Extension Methods (C# Programming Guide).

Remarks

The left null space of a matrix is the set of vectors that are orthogonal to the columns of the matrix. It is the (right) null space of the transpose of the matrix.

This method returns an orthogonal (unitary) matrix whose columns form a basis for this set. Multiplying the transpose of this matrix with the matrix results in the null vector.

The null space is computed using the singular value decomposition of the matrix. The rank of the matrix is determined by counting the number of singular values whose absolute value is greater than the tolerance. The tolerance is equal to the product of the machine precision, the largest dimension of the matrix, and the spectral norm (largest singular value) of the matrix.

GetLeftNullSpace<T>(Matrix<T>, T)

Computes an orthogonal basis for the left null space of a matrix.
C# 
public static Matrix<T> GetLeftNullSpace<T>(
	this Matrix<T> matrix,
	T tolerance
)

Parameters

matrix  Matrix<T>
The matrix whose left null space is to be computed.
tolerance  T
The tolerance used to determine the rank of the matrix.

Type Parameters

T
The type of the elements of the matrix.

Return Value

Matrix<T>
A matrix representing the left null space of the input matrix.

Usage Note

In Visual Basic and C#, you can call this method as an instance method on any object of type Matrix<T>. When you use instance method syntax to call this method, omit the first parameter. For more information, see Extension Methods (Visual Basic) or Extension Methods (C# Programming Guide).

Remarks

The left null space of a matrix is the set of vectors that are orthogonal to the columns of the matrix. It is the (right) null space of the transpose of the matrix.

This method returns an orthogonal (unitary) matrix whose columns form a basis for this set. Multiplying the transpose of this matrix with the matrix results in the null vector.

The null space is computed using the singular value decomposition of the matrix. The rank of the matrix is determined by counting the number of singular values whose absolute value is greater than the tolerance.

If tolerance is negative, the default tolerance is used. The default tolerance is equal to the product of the machine precision, the largest dimension of the matrix, and the spectral norm (largest singular value) of the matrix.

See Also