RealEigenvalueDecomposition<T> Class

Represents the eigenvalue decomposition of a non-symmetric, real matrix.

Definition

Namespace: Numerics.NET.LinearAlgebra
Assembly: Numerics.NET (in Numerics.NET.dll) Version: 9.0.2
C#
public abstract class RealEigenvalueDecomposition<T> : EigenvalueDecomposition<T>
Inheritance
Object  →  LinearOperator<T>  →  Decomposition<T>  →  EigenvalueDecomposition<T>  →  RealEigenvalueDecomposition<T>

Type Parameters

T

Remarks

Use the RealEigenvalueDecomposition<T> class to get access to properties specific to the eigenvalue decomposition of a real, square matrix.

The eigenvalue decomposition of a real square general matrix A rewrites the matrix in the form A = XLX-1, where X is a matrix and L is a diagonal matrix. The diagonal elements of L are the eigenvalues. The columns of X are the corresponding eigenvectors.

The eigenvalues are either real or come in complex conjugate pairs. The eigenvectors corresponding to real eigenvalues are also real. The eigenvectors corresponding to pairs of complex conjugate eigenvalues are themselves complex conjugates. The Eigenvalues and Eigenvectors properties return only the real eigenvalues and eigenvectors. The ComplexEigenvalues and ComplexEigenvectors properties return the full set of eigenvalues and eigenvectors. The RawEigenvectors property returns the real and imaginary parts of all eigenvectors.

The eigenvalues of a real symmetric matrix are always real, and its eigenvectors are real and orthogonal. Its eigenvalue decomposition can be calculated more easily, and there is no need to allow for complex eigenvectors and eigenvalues.

EigenvalueDecomposition<T> inherits from LinearOperator<T>. However, unlike the other decompositions, the non-symmetric eigenvalue decomposition does not offer a speed advantage when performing calculations most calculations. For example, solving an equation using the eigen decomposition itself requires the solution of a system of equations. For this reason, the Solve(Vector<T>), GetInverse(Boolean) and EstimateConditionNumber() methods simply call the corresponding method on the base matrix.

Constructors

Properties

BaseMatrix Gets the underlying matrix of the decomposition.
(Inherited from Decomposition<T>)
ColumnCount Gets the number of columns in the matrix.
(Inherited from LinearOperator<T>)
ColumnIndex Gets or sets the index of labels for the columns of the matrix.
(Inherited from LinearOperator<T>)
Preliminary
ComplexEigenvalues Gets the eigenvalues of the matrix as a complex vector.
(Inherited from EigenvalueDecomposition<T>)
ComplexEigenvectors Gets the eigenvectors of the matrix as a complex matrix.
(Inherited from EigenvalueDecomposition<T>)
Done Gets or sets a value that indicates whether the decomposition has been performed.
(Inherited from Decomposition<T>)
Eigenvalues Gets the eigenvalues of the matrix.
(Inherited from EigenvalueDecomposition<T>)
Eigenvectors Gets the eigenvectors of the matrix.
(Inherited from EigenvalueDecomposition<T>)
ElementType Gets the element type of the matrix.
(Inherited from LinearOperator<T>)
Factors Gets a collection of the matrix factors that make up the decomposition.
(Inherited from EigenvalueDecomposition<T>)
HasComplexEigenvalues Indicates whether the matrix has complex eigenvalue.
(Inherited from EigenvalueDecomposition<T>)
Overwrite Gets or sets whether the BaseMatrix should be overwritten by its decomposition.
(Inherited from Decomposition<T>)
RawEigenvectors Gets the eigenvectors of the matrix.
(Inherited from EigenvalueDecomposition<T>)
RowCount Gets the number of rows in the matrix.
(Inherited from LinearOperator<T>)
RowIndex Gets or sets the index of labels for the rows of the matrix.
(Inherited from LinearOperator<T>)
Preliminary

Methods

Decompose Performs the actual decomposition.
(Inherited from Decomposition<T>)
EqualsDetermines whether the specified object is equal to the current object.
(Inherited from Object)
EstimateConditionNumber Calculates an estimate for the condition number of a matrix.
(Inherited from LinearOperator<T>)
FinalizeAllows an object to try to free resources and perform other cleanup operations before it is reclaimed by garbage collection.
(Inherited from Object)
GetDeterminant Calculates the determinant of the decomposed matrix.
(Overrides EigenvalueDecomposition<T>.GetDeterminant())
GetHashCodeServes as the default hash function.
(Inherited from Object)
GetInverse() Calculates the inverse matrix..
(Inherited from LinearOperator<T>)
GetInverse(Boolean) Calculates the inverse of the factorized matrix.
(Inherited from EigenvalueDecomposition<T>)
GetObjectData Populates a SerializationInfo with the data needed to serialize the target object.
(Inherited from LinearOperator<T>)
Obsolete.
GetTypeGets the Type of the current instance.
(Inherited from Object)
IsSingular Gets whether the BaseMatrix of this decomposition is singular.
(Inherited from Decomposition<T>)
LeastSquaresSolve(Matrix<T>) Solves the system of linear equations for the specified right-hand side matrix in the least squares sense.
(Inherited from LinearOperator<T>)
LeastSquaresSolve(Vector<T>) Solves the system of linear equations for the specified right-hand side vector.
(Inherited from LinearOperator<T>)
LeastSquaresSolve(DenseMatrix<T>, Boolean) Solves the system of linear equations for the specified right-hand side dense matrix in the least squares sense and optionally overwrites the right-hand side with the solution.
(Inherited from LinearOperator<T>)
LeastSquaresSolve(DenseVector<T>, Boolean) Solves the system of linear equations for the specified right-hand side dense vector in the least squares sense and optionally overwrites the right-hand side with the solution.
(Inherited from LinearOperator<T>)
LeastSquaresSolveInto(Matrix<T>, Matrix<T>) Solves a system of equations defined by the matrix and multiple right-hand sides in the least squares sense.
(Inherited from LinearOperator<T>)
LeastSquaresSolveInto(Vector<T>, Vector<T>) Solves a system of equations defined by the matrix and the specified right-hand side in the least squares sense.
(Inherited from Decomposition<T>)
MemberwiseCloneCreates a shallow copy of the current Object.
(Inherited from Object)
Multiply(Matrix<T>)Applies the linear operator to a matrix.
(Inherited from LinearOperator<T>)
Multiply(Vector<T>)Applies the linear operator to a vector.
(Inherited from LinearOperator<T>)
MultiplyAndAddAsLeftFactorCore Multiplies the matrix with a vector, adds the scaled result to another scaled vector, and returns the result.
(Inherited from Decomposition<T>)
MultiplyAsLeftCore(TransposeOperation, Matrix<T>, Matrix<T>) Multiplies a matrix by the linear operator and returns the result.
(Inherited from Decomposition<T>)
MultiplyAsLeftCore(TransposeOperation, Vector<T>, Vector<T>) Multiplies a vector by the linear operator and returns the result.
(Inherited from Decomposition<T>)
MultiplyTranspose(Matrix<T>)Applies the transpose of the linear operator to a matrix.
(Inherited from LinearOperator<T>)
MultiplyTranspose(Vector<T>)Applies the (conjugate) transpose of the linear operator to a vector.
(Inherited from LinearOperator<T>)
Rank() Returns the numerical rank of a matrix.
(Inherited from LinearOperator<T>)
Rank(T) Returns the numerical rank of a matrix using the specified tolerance.
(Overrides EigenvalueDecomposition<T>.Rank(T))
SetSingular Sets a flag that indicates the underlying matrix of this decomposition is singular.
(Inherited from Decomposition<T>)
Solve(Matrix<T>) Solves the system of linear equations for the specified right-hand side matrix.
(Inherited from LinearOperator<T>)
Solve(Vector<T>) Solves the system of linear equations for the specified right-hand side vector.
(Inherited from LinearOperator<T>)
Solve(DenseMatrix<T>, Boolean) Solves the system of linear equations for the specified right-hand side dense matrix and optionally overwrites the right-hand side with the solution.
(Inherited from LinearOperator<T>)
Solve(DenseVector<T>, Boolean) Solves the system of linear equations for the specified right-hand side dense vector and optionally overwrites the right-hand side with the solution.
(Inherited from LinearOperator<T>)
SolveInto(Matrix<T>, Matrix<T>) Solves a system of equations defined by the matrix and multiple right-hand sides.
(Inherited from LinearOperator<T>)
SolveInto(Vector<T>, Vector<T>) Solves a system of equations defined by the matrix and the specified right-hand side.
(Inherited from LinearOperator<T>)
SolveInto(TransposeOperation, Matrix<T>, Matrix<T>) Solves a system of equations defined by the matrix and multiple right-hand sides.
(Inherited from EigenvalueDecomposition<T>)
SolveInto(TransposeOperation, Vector<T>, Vector<T>) Solves a system of equations defined by the matrix and the specified right-hand side.
(Inherited from EigenvalueDecomposition<T>)
SolveTranspose(Matrix<T>) Solves the transposed system of linear equations for the specified right-hand side matrix.
(Inherited from LinearOperator<T>)
SolveTranspose(Vector<T>) Solves the system of linear equations for the specified right-hand side vector.
(Inherited from LinearOperator<T>)
SolveTranspose(DenseMatrix<T>, Boolean) Solves the transposed system of linear equations for the specified right-hand side dense matrix and optionally overwrites the right-hand side with the solution.
(Inherited from LinearOperator<T>)
SolveTranspose(DenseVector<T>, Boolean) Solves the transposed system of linear equations for the specified right-hand side dense vector and optionally overwrites the right-hand side with the solution.
(Inherited from LinearOperator<T>)
SolveTransposeInto(Matrix<T>, Matrix<T>) Solves a system of equations defined by the matrix and multiple right-hand sides.
(Inherited from LinearOperator<T>)
SolveTransposeInto(Vector<T>, Vector<T>) Solves a system of equations defined by the matrix and the specified right-hand side.
(Inherited from LinearOperator<T>)
ToStringReturns a string that represents the current object.
(Inherited from Object)

Extension Methods

EstimateConditionNumber<T> Calculates an estimate for the condition number of the matrix.
(Defined by MatrixExtensions)

See Also