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ComplexEigenvalueDecomposition ClassExtreme Optimization Numerical Libraries for .NET Professional
Represents the eigenvalue decomposition of a dense matrix.
Inheritance Hierarchy

SystemObject
  Extreme.Mathematics.LinearAlgebra.ComplexComplexLinearOperator
    Extreme.Mathematics.LinearAlgebra.ComplexComplexDecomposition
      Extreme.Mathematics.LinearAlgebraComplexEigenvalueDecomposition
        Extreme.Mathematics.LinearAlgebraHermitianEigenvalueDecomposition
        Extreme.Mathematics.LinearAlgebraNonHermitianEigenvalueDecomposition

Namespace: Extreme.Mathematics.LinearAlgebra
Assembly: Extreme.Numerics.Net40 (in Extreme.Numerics.Net40.dll) Version: 6.0.16073.0 (6.0.17114.0)
Syntax

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public abstract class ComplexEigenvalueDecomposition : ComplexDecomposition
Public MustInherit Class ComplexEigenvalueDecomposition
	Inherits ComplexDecomposition
public ref class ComplexEigenvalueDecomposition abstract : public ComplexDecomposition
[<AbstractClassAttribute>]
type ComplexEigenvalueDecomposition =  
    class
        inherit ComplexDecomposition
    end

The ComplexEigenvalueDecomposition type exposes the following members.

Constructors

  NameDescription
Public methodComplexEigenvalueDecomposition
Constructs a new EigenvalueDecomposition object.
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Properties

  NameDescription
Public propertyBaseMatrix
Gets the underlying matrix of the decomposition.
(Inherited from ComplexDecomposition.)
Protected propertyDone
Gets or sets whether the decomposition has been performed.
(Inherited from ComplexDecomposition.)
Public propertyEigenvalues
Gets the eigenvalues of the matrix.
Public propertyEigenvectors
Gets the eigenvectors of the matrix as a complex matrix.
Public propertyFactors
Gets a collection of the matrix factors that make up the decomposition.
(Overrides ComplexDecompositionFactors.)
Public propertyOverwrite
Gets or sets a value indicating whether the BaseMatrix should be overwritten by its decomposition.
(Inherited from ComplexDecomposition.)
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Methods

  NameDescription
Public methodDecompose
Performs the actual decomposition.
(Inherited from ComplexDecomposition.)
Public methodEquals
Determines whether the specified Object is equal to the current Object.
(Inherited from Object.)
Public methodEstimateConditionNumber
Calculates an estimate for the condition number of the matrix.
(Overrides ComplexLinearOperatorEstimateConditionNumber.)
Protected methodFinalize
Allows an object to try to free resources and perform other cleanup operations before it is reclaimed by garbage collection.
(Inherited from Object.)
Public methodGetDeterminant
Calculates the determinant of the decomposed matrix.
(Overrides ComplexLinearOperatorGetDeterminant.)
Public methodGetHashCode
Serves as a hash function for a particular type.
(Inherited from Object.)
Public methodGetInverse
Calculates the inverse matrix..
(Inherited from ComplexLinearOperator.)
Public methodGetInverse(Boolean)
Calculates the inverse of the factorized matrix.
(Overrides ComplexLinearOperatorGetInverse(Boolean).)
Public methodGetObjectData
Populates a SerializationInfo with the data needed to serialize the target object.
(Inherited from ComplexLinearOperator.)
Public methodGetType
Gets the Type of the current instance.
(Inherited from Object.)
Public methodIsSingular
Gets a value indicating whether the BaseMatrix of this decomposition is singular.
(Inherited from ComplexDecomposition.)
Protected methodMemberwiseClone
Creates a shallow copy of the current Object.
(Inherited from Object.)
Public methodMultiply(ComplexMatrix)
Multiplies the decomposition to a matrix.
(Inherited from ComplexDecomposition.)
Public methodMultiply(ComplexVector)
Applies the base matrix of the decomposition to a vector.
(Inherited from ComplexDecomposition.)
Public methodMultiplyTranspose(ComplexMatrix)
Multiplies the transpose of the decomposition to a matrix.
(Inherited from ComplexDecomposition.)
Public methodMultiplyTranspose(ComplexVector)
Multiplies the transpose of the decomposition to a vector.
(Inherited from ComplexDecomposition.)
Public methodRank
Returns the numerical rank of a matrix.
(Inherited from ComplexLinearOperator.)
Public methodRank(Double)
Returns the numerical rank of a matrix using the specified tolerance.
(Overrides ComplexLinearOperatorRank(Double).)
Protected methodSetSingular
Sets a flag that indicates the underlying matrix of this decomposition is singular.
(Inherited from ComplexDecomposition.)
Public methodSolve(ComplexMatrix)
Solves the system of linear equations for the specified right-hand side ComplexMatrix.
(Inherited from ComplexLinearOperator.)
Public methodSolve(ComplexVector)
Solves the system of linear equations for the specified right-hand side complex vector.
(Inherited from ComplexLinearOperator.)
Public methodSolve(ComplexDenseMatrix, Boolean)
Solves the system of simultaneous linear equations for the specified right-hand side dense matrix and overwrites the right-hand side with the solution.
(Overrides ComplexLinearOperatorSolve(ComplexDenseMatrix, Boolean).)
Public methodSolve(ComplexDenseVector, Boolean)
Solves the system of simultaneous linear equations for the specified right-hand side vector.
(Overrides ComplexLinearOperatorSolve(ComplexDenseVector, Boolean).)
Public methodSolveTranspose(ComplexMatrix)
Solves the transposed system of linear equations for the specified right-hand side ComplexMatrix.
(Inherited from ComplexLinearOperator.)
Public methodSolveTranspose(ComplexVector)
Solves the system of linear equations for the specified right-hand side complex vector.
(Inherited from ComplexLinearOperator.)
Public methodSolveTranspose(ComplexDenseMatrix, 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.
(Overrides ComplexLinearOperatorSolveTranspose(ComplexDenseMatrix, Boolean).)
Public methodSolveTranspose(ComplexDenseVector, 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.
(Overrides ComplexLinearOperatorSolveTranspose(ComplexDenseVector, Boolean).)
Public methodToString
Returns a string that represents the current object.
(Inherited from Object.)
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Remarks

The eigenvalue decomposition of a complex 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 complex or come in complex conjugate pairs. The eigenvectors corresponding to complex eigenvalues are also complex. The eigenvectors corresponding to pairs of complex conjugate eigenvalues are themselves complex conjugates.

The eigenvalues of a complex symmetric matrix are always complex, and its eigenvectors are orthogonal. Its eigenvalue decomposition can be calculated more easily. This is implemented by the SymmetricEigenvalueDecomposition class.

EigenvalueDecomposition inherits from LinearOperator. However, unlike the other decompositions, the nonsymmetric 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(ComplexDenseMatrix, Boolean), GetInverse(Boolean) and EstimateConditionNumber methods simply call the corresponding method on the base matrix.

Version Information

Numerical Libraries

Supported in: 5.x
See Also

Reference

Extreme.Mathematics.LinearAlgebra Namespace
Extreme.Mathematics.LinearAlgebraDenseMatrix
Extreme.Mathematics.LinearAlgebraSymmetricEigenvalueDecomposition
Extreme.Mathematics.LinearAlgebraLinearOperator

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