GeneralizedEigenvalueDecompositionT Class

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.

	Name	Description
	GeneralizedEigenvalueDecompositionT	Constructs a new GeneralizedEigenvalueDecompositionT object.

	Name	Description
	BaseMatrix	Gets the primary underlying matrix of the decomposition. (Inherited from GeneralizedDecompositionT.)
	ComplexEigenvalueNumerators	Gets the numerators of the eigenvalues as a complex vector.
	ComplexEigenvalues	Gets the eigenvalues as a complex vector.
	ComplexEigenvectors	Gets the eigenvectors as a complex matrix.
	Done	Gets or sets a value that indicates whether the decomposition has been performed. (Inherited from GeneralizedDecompositionT.)
	EigenvalueDenominators	Gets the denominators of the eigenvalues.
	EigenvalueNumerators	Gets the numerators of the eigenvalues.
	Eigenvalues	Gets the eigenvalues.
	Eigenvectors	Gets the eigenvectors.
	HasComplexEigenvalues	Indicates whether the matrix has complex eigenvalue.
	Overwrite	Gets or sets a value indicating whether the BaseMatrix should be overwritten by its decomposition. (Inherited from GeneralizedDecompositionT.)
	RawEigenvectors	Gets the eigenvectors.
	SecondaryBaseMatrix	Gets the secondary underlying matrix of the decomposition. (Inherited from GeneralizedDecompositionT.)

	Name	Description
	Decompose	Performs the actual decomposition. (Inherited from GeneralizedDecompositionT.)
	Equals	Determines whether the specified object is equal to the current object. (Inherited from Object.)
	Finalize	Allows an object to try to free resources and perform other cleanup operations before it is reclaimed by garbage collection. (Inherited from Object.)
	GetHashCode	Serves as the default hash function. (Inherited from Object.)
	GetType	Gets the Type of the current instance. (Inherited from Object.)
	MemberwiseClone	Creates a shallow copy of the current Object. (Inherited from Object.)
	SetSingular	Sets a flag that indicates the underlying matrix of this decomposition is singular. (Inherited from GeneralizedDecompositionT.)
	ToString	Returns a string that represents the current object. (Inherited from Object.)

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.

Reference