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  • Extreme.Mathematics.LinearAlgebra
    • BandMatrix(T) Class
    • BlockVector(T) Class
    • CholeskyDecomposition(T) Class
    • CloningMethod Enumeration
    • ColumnCollection(T) Structure
    • ComplexSingularValueDecomposition(T) Class
    • ComponentReadOnlyException Class
    • ComposedComplexMatrix(T) Class
    • ComposedComplexVector(T) Class
    • ConstantMatrix(T) Class
    • ConstantVector(T) Class
    • Decomposition(T) Class
    • DenseMatrix(T) Class
    • DenseVector(T) Class
    • DiagonalMatrix(T) Class
    • EigenvalueDecomposition(T) Class
    • EigenvalueRange Enumeration
    • GeneralizedDecomposition(T) Class
    • GeneralizedEigenvalueDecomposition(T) Class
    • GeneralizedSingularValueDecomposition(T) Class
    • GeneralizedSingularValueDecompositionFactors Enumeration
    • HermitianMatrix(T) Class
    • IndexedVector(T) Class
    • IndexValuePair(T) Structure
    • IResizableMatrix(T) Interface
    • LeastSquaresSolutionMethod Enumeration
    • LeastSquaresSolver(T) Class
    • LinearAlgebraOperations Class
    • LinearOperator(T) Class
    • LQDecomposition(T) Class
    • LUDecomposition(T) Class
    • MatrixNotPositiveDefiniteException Class
    • MatrixSingularException Class
    • MatrixView(T) Class
    • NonHermitianEigenvalueDecomposition(T) Class
    • NonNegativeMatrixFactorization(T) Class
    • PermutationMatrix Class
    • PivotVector Structure
    • QLDecomposition(T) Class
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    • RealEigenvalueDecomposition(T) Class
    • RowCollection(T) Structure
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    • SparseCompressedColumnMatrix(T) Class
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    • SymmetricIndefiniteDecomposition(T) Class
    • SymmetricMatrix(T) Class
    • TriangularMatrix(T) Class
  • GeneralizedEigenvalueDecomposition(T) Class
    • GeneralizedEigenvalueDecomposition(T) Constructor
    • Properties
    • GeneralizedEigenvalueDecomposition(T) Methods

GeneralizedEigenvalueDecompositionT Class

Extreme Optimization Numerical Libraries for .NET Professional
Represents the generalized eigenvalue decomposition of two matrices.
Inheritance Hierarchy

SystemObject
  Extreme.Mathematics.LinearAlgebraGeneralizedDecompositionT
    Extreme.Mathematics.LinearAlgebraGeneralizedEigenvalueDecompositionT

Namespace:  Extreme.Mathematics.LinearAlgebra
Assembly:  Extreme.Numerics (in Extreme.Numerics.dll) Version: 8.1.1
Syntax

C#
VB
C++
F#
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public abstract class GeneralizedEigenvalueDecomposition<T> : GeneralizedDecomposition<T>
Public MustInherit Class GeneralizedEigenvalueDecomposition(Of T)
	Inherits GeneralizedDecomposition(Of T)
generic<typename T>
public ref class GeneralizedEigenvalueDecomposition abstract : public GeneralizedDecomposition<T>
[<AbstractClassAttribute>]
type GeneralizedEigenvalueDecomposition<'T> =  
    class
        inherit GeneralizedDecomposition<'T>
    end

Type Parameters

T

The GeneralizedEigenvalueDecompositionT type exposes the following members.

Constructors

  NameDescription
Protected methodGeneralizedEigenvalueDecompositionT
Constructs a new GeneralizedEigenvalueDecompositionT object.
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Properties

  NameDescription
Public propertyBaseMatrix
Gets the primary underlying matrix of the decomposition.
(Inherited from GeneralizedDecompositionT.)
Public propertyComplexEigenvalueNumerators
Gets the numerators of the eigenvalues as a complex vector.
Public propertyComplexEigenvalues
Gets the eigenvalues as a complex vector.
Public propertyComplexEigenvectors
Gets the eigenvectors as a complex matrix.
Protected propertyDone
Gets or sets a value that indicates whether the decomposition has been performed.
(Inherited from GeneralizedDecompositionT.)
Public propertyEigenvalueDenominators
Gets the denominators of the eigenvalues.
Public propertyEigenvalueNumerators
Gets the numerators of the eigenvalues.
Public propertyEigenvalues
Gets the eigenvalues.
Public propertyEigenvectors
Gets the eigenvectors.
Public propertyHasComplexEigenvalues
Indicates whether the matrix has complex eigenvalue.
Public propertyOverwrite
Gets or sets a value indicating whether the BaseMatrix should be overwritten by its decomposition.
(Inherited from GeneralizedDecompositionT.)
Public propertyRawEigenvectors
Gets the eigenvectors.
Public propertySecondaryBaseMatrix
Gets the secondary underlying matrix of the decomposition.
(Inherited from GeneralizedDecompositionT.)
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Methods

  NameDescription
Public methodDecompose
Performs the actual decomposition.
(Inherited from GeneralizedDecompositionT.)
Public methodEquals
Determines whether the specified object is equal to the current object.
(Inherited from Object.)
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 methodGetHashCode
Serves as the default hash function.
(Inherited from Object.)
Public methodGetType
Gets the Type of the current instance.
(Inherited from Object.)
Protected methodMemberwiseClone
Creates a shallow copy of the current Object.
(Inherited from Object.)
Protected methodSetSingular
Sets a flag that indicates the underlying matrix of this decomposition is singular.
(Inherited from GeneralizedDecompositionT.)
Public methodToString
Returns a string that represents the current object.
(Inherited from Object.)
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Remarks

The eigenvalue decomposition of a pair of square matrices A and B computes scalars ?,? and vectors x, y so that

Ax = ?Bx

.

or

.

?Ay = By

.

The scalars are called (generalized) eigenvalues and the vectors are called eigenvectors. If neither ? nor ? is zero, then the two problems are equivalent with x=y and ?=1/?. To cover the case where either may be zero or very small, two sets of values are returned, so that the eigenvalues are equal to their quotient.

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 generalized eigenvalues of a pair of real symmetric or complex Hermitian matrices where the secondary matrix is positive definite are always real,and its eigenvectors are orthogonal with respect to the second matrix. Its eigenvalue decomposition can be calculated more easily.

GeneralizedEigenvalueDecompositionT inherits fromGeneralizedDecompositionT.

See Also

Reference

Extreme.Mathematics.LinearAlgebra Namespace
Extreme.Mathematics.LinearAlgebraEigenvalueDecompositionT
Extreme.Mathematics.LinearAlgebraGeneralizedDecompositionT

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