Extreme Optimization™: Complexity made simple.

Math and Statistics
Libraries for .NET

  • Home
  • Features
    • Math Library
    • Vector and Matrix Library
    • Statistics Library
    • Performance
    • Usability
  • Documentation
    • Introduction
    • Math Library User's Guide
    • Vector and Matrix Library User's Guide
    • Data Analysis Library User's Guide
    • Statistics Library User's Guide
    • Reference
  • Resources
    • Downloads
    • QuickStart Samples
    • Sample Applications
    • Frequently Asked Questions
    • Technical Support
  • Blog
  • Order
  • Company
    • About us
    • Testimonials
    • Customers
    • Press Releases
    • Careers
    • Partners
    • Contact us
Introduction
Deployment Guide
Configuration
Using Parallelism
Expand Mathematics Library User's GuideMathematics Library User's Guide
Expand Vector and Matrix Library User's GuideVector and Matrix Library User's Guide
Expand Data Analysis Library User's GuideData Analysis Library User's Guide
Expand Statistics Library User's GuideStatistics Library User's Guide
Expand Data Access Library User's GuideData Access Library User's Guide
Expand ReferenceReference
  • Extreme Optimization
    • Features
    • Solutions
    • Documentation
    • QuickStart Samples
    • Sample Applications
    • Downloads
    • Technical Support
    • Download trial
    • How to buy
    • Blog
    • Company
    • Resources
  • Documentation
    • Introduction
    • Deployment Guide
    • Configuration
    • Using Parallelism
    • Mathematics Library User's Guide
    • Vector and Matrix Library User's Guide
    • Data Analysis Library User's Guide
    • Statistics Library User's Guide
    • Data Access Library User's Guide
    • Reference
  • Reference
    • Extreme
    • Extreme.Collections
    • Extreme.Data
    • Extreme.Data.Json
    • Extreme.Data.Matlab
    • Extreme.Data.R
    • Extreme.Data.Stata
    • Extreme.Data.Text
    • Extreme.DataAnalysis
    • Extreme.DataAnalysis.Linq
    • Extreme.Mathematics
    • Extreme.Mathematics.Algorithms
    • Extreme.Mathematics.Calculus
    • Extreme.Mathematics.Calculus.OrdinaryDifferentialEquations
    • Extreme.Mathematics.Curves
    • Extreme.Mathematics.Curves.Nonlinear
    • Extreme.Mathematics.Distributed
    • Extreme.Mathematics.Distributed.Cuda
    • Extreme.Mathematics.EquationSolvers
    • Extreme.Mathematics.FSharp
    • Extreme.Mathematics.Generic
    • Extreme.Mathematics.Generic.LinearAlgebra
    • Extreme.Mathematics.Generic.LinearAlgebra.Implementation
    • Extreme.Mathematics.Generic.LinearAlgebra.Providers
    • Extreme.Mathematics.Generic.SignalProcessing
    • Extreme.Mathematics.Implementation
    • Extreme.Mathematics.LinearAlgebra
    • Extreme.Mathematics.LinearAlgebra.Complex
    • Extreme.Mathematics.LinearAlgebra.Complex.Decompositions
    • Extreme.Mathematics.LinearAlgebra.Implementation
    • Extreme.Mathematics.LinearAlgebra.IO
    • Extreme.Mathematics.LinearAlgebra.IterativeSolvers
    • Extreme.Mathematics.LinearAlgebra.IterativeSolvers.Preconditioners
    • Extreme.Mathematics.LinearAlgebra.Providers
    • Extreme.Mathematics.LinearAlgebra.Sparse
    • Extreme.Mathematics.Optimization
    • Extreme.Mathematics.Optimization.Genetic
    • Extreme.Mathematics.Optimization.LineSearches
    • Extreme.Mathematics.Random
    • Extreme.Mathematics.SignalProcessing
    • Extreme.Numerics.FSharp
    • Extreme.Statistics
    • Extreme.Statistics.Distributions
    • Extreme.Statistics.IO
    • Extreme.Statistics.Linq
    • Extreme.Statistics.Multivariate
    • Extreme.Statistics.Random
    • Extreme.Statistics.Tests
    • Extreme.Statistics.TimeSeriesAnalysis
  • Extreme.Mathematics.LinearAlgebra
    • BandMatrix Class
    • BandMatrix(T) Class
    • BandVector Class
    • BlockVector(T) Class
    • CholeskyDecomposition Class
    • CholeskyDecomposition(T) Class
    • CloningMethod Enumeration
    • ColumnCollection Structure
    • ColumnCollection(T) Structure
    • ComplexEigenvalueDecomposition Class
    • ComplexSingularValueDecomposition(T) Class
    • ComponentReadOnlyException Class
    • ComposedComplexMatrix(T) Class
    • ComposedComplexVector(T) Class
    • ConstantMatrix Class
    • ConstantMatrix(T) Class
    • ConstantVector Class
    • ConstantVector(T) Class
    • Decomposition Class
    • Decomposition(T) Class
    • DenseMatrix Class
    • DenseMatrix(T) Class
    • DenseVector Class
    • DenseVector(T) Class
    • DiagonalMatrix Class
    • DiagonalMatrix(T) Class
    • EigenvalueDecomposition Class
    • EigenvalueDecomposition(T) Class
    • EigenvalueRange Enumeration
    • GeneralizedDecomposition(T) Class
    • GeneralizedEigenvalueDecomposition(T) Class
    • GeneralizedSingularValueDecomposition(T) Class
    • GeneralizedSingularValueDecompositionFactors Enumeration
    • HermitianEigenvalueDecomposition Class
    • HermitianMatrix(T) Class
    • IDoubleStorage Interface
    • ILeastSquaresSolver Interface
    • IndexedVector Class
    • IndexedVector(T) Class
    • IndexValuePair Structure
    • IndexValuePair(T) Structure
    • IResizableMatrix Interface
    • IResizableMatrix(T) Interface
    • ISingleLeastSquaresSolver Interface
    • ISingleResizableMatrix Interface
    • ISingleStorage Interface
    • Kernel Class
    • LeastSquaresSolutionMethod Enumeration
    • LeastSquaresSolver Class
    • LeastSquaresSolver(T) Class
    • LinearAlgebraOperations Class
    • LinearOperator Class
    • LinearOperator(T) Class
    • LQDecomposition(T) Class
    • LUDecomposition Class
    • LUDecomposition(T) Class
    • MatrixNotPositiveDefiniteException Class
    • MatrixSingularException Class
    • MatrixStructure Enumeration
    • MatrixView Class
    • MatrixView(T) Class
    • NonHermitianEigenvalueDecomposition Class
    • NonHermitianEigenvalueDecomposition(T) Class
    • NonNegativeMatrixFactorization Class
    • NonNegativeMatrixFactorization(T) Class
    • NonsymmetricEigenvalueDecomposition Class
    • PermutationMatrix Class
    • PivotVector Structure
    • QLDecomposition(T) Class
    • QRDecomposition Class
    • QRDecomposition(T) Class
    • Range Structure
    • RowCollection Structure
    • RowCollection(T) Structure
    • RowColumnValueTriplet Structure
    • RowColumnValueTriplet(T) Structure
    • RQDecomposition(T) Class
    • SingleBandCholeskyDecomposition Class
    • SingleBandLUDecomposition Class
    • SingleBandMatrix Class
    • SingleBandVector Class
    • SingleBandVector.BandVectorEnumerator Structure
    • SingleCholeskyDecomposition Class
    • SingleColumnCollection Structure
    • SingleConstantMatrix Class
    • SingleConstantVector Class
    • SingleDecomposition Class
    • SingleDenseMatrix Class
    • SingleDenseVector Class
    • SingleDiagonalMatrix Class
    • SingleEigenvalueDecomposition Class
    • SingleIndexedVector Class
    • SingleIndexValuePair Structure
    • SingleKernel Class
    • SingleLeastSquaresSolver Class
    • SingleLinearTransformation Class
    • SingleLUDecomposition Class
    • SingleMatrixView Class
    • SinglePermutationMatrix Class
    • SingleQRDecomposition Class
    • SingleRowCollection Structure
    • SingleRowColumnValueTriplet Structure
    • SingleSingularValueDecomposition Class
    • SingleSparseLUDecomposition Class
    • SingleSparseMatrix Class
    • SingleSparseVector Class
    • SingleSymmetricEigenvalueDecomposition Class
    • SingleSymmetricMatrix Class
    • SingleTriangularMatrix Class
    • SingularValueDecomposition Class
    • SingularValueDecomposition(T) Class
    • SingularValueDecompositionFactors Enumeration
    • SparseCompressedColumnMatrix Class
    • SparseCompressedColumnMatrix(T) Class
    • SparseMatrix Class
    • SparseMatrix(T) Class
    • SparseVector Class
    • SparseVector(T) Class
    • SymmetricEigenvalueDecomposition Class
    • SymmetricIndefiniteDecomposition Class
    • SymmetricIndefiniteDecomposition(T) Class
    • SymmetricMatrix Class
    • SymmetricMatrix(T) Class
    • TriangularMatrix Class
    • TriangularMatrix(T) Class
  • ComplexEigenvalueDecomposition Class
    • ComplexEigenvalueDecomposition Constructor
    • Properties
    • Methods
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

C#
VB
C++
F#
Copy
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.
Top
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.)
Top
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.)
Top
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

Copyright (c) 2004-2017 ExoAnalytics Inc.

Send comments on this topic to support@extremeoptimization.com

Copyright © 2004-2018, Extreme Optimization. All rights reserved.
Extreme Optimization, Complexity made simple, M#, and M Sharp are trademarks of ExoAnalytics Inc.
Microsoft, Visual C#, Visual Basic, Visual Studio, Visual Studio.NET, and the Optimized for Visual Studio logo
are registered trademarks of Microsoft Corporation.