public abstract class EigenvalueDecomposition : Decomposition

Public MustInherit Class EigenvalueDecomposition
	Inherits Decomposition

public ref class EigenvalueDecomposition abstract : public Decomposition

[<AbstractClassAttribute>]
type EigenvalueDecomposition =  
    class
        inherit Decomposition
    end

The EigenvalueDecomposition type exposes the following members.

	Name	Description
	EigenvalueDecomposition	Constructs a new EigenvalueDecomposition object.

	Name	Description
	BaseMatrix	Gets the underlying matrix of the decomposition. (Inherited from Decomposition.)
	ColumnCount	Gets the number of columns in the matrix. (Inherited from Decomposition.)
	ComplexEigenvalues	Gets the eigenvalues of the matrix as a complex vector.
	ComplexEigenvectors	Gets the eigenvectors of the matrix as a complex matrix.
	Done	Gets or sets whether the decomposition has been performed. (Inherited from Decomposition.)
	Factors	Gets a collection of the matrix factors that make up the decomposition. (Overrides DecompositionFactors.)
	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 Decomposition.)
	RawEigenvectors	Gets the eigenvectors of the matrix.
	RealEigenvalues	Gets the eigenvalues of the matrix.
	RealEigenvectors	Gets the real eigenvectors of the matrix.
	RowCount	Gets the number of rows in the matrix. (Inherited from Decomposition.)

	Name	Description
	AddProductToVector	Multiplies a Vector by this Matrix and stores the result in a second vector of type DenseVector. (Inherited from LinearOperator.)
	Decompose	Performs the actual decomposition. (Inherited from Decomposition.)
	Equals	Determines whether the specified Object is equal to the current Object. (Inherited from Object.)
	EstimateConditionNumber	Calculates an estimate for the condition number of the matrix. (Overrides LinearOperatorEstimateConditionNumber.)
	Finalize	Allows 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 LinearOperatorGetDeterminant.)
	GetHashCode	Serves as a hash function for a particular type. (Inherited from Object.)
	GetInverse	Calculates the inverse matrix.. (Inherited from LinearOperator.)
	GetInverse(Boolean)	Calculates the inverse of the factorized matrix. (Overrides LinearOperatorGetInverse(Boolean).)
	GetObjectData	Populates a SerializationInfo with the data needed to serialize the target object. (Inherited from LinearOperator.)
	GetType	Gets the Type of the current instance. (Inherited from Object.)
	IsSingular	Gets a value indicating whether the BaseMatrix of this decomposition is singular. (Inherited from Decomposition.)
	MemberwiseClone	Creates a shallow copy of the current Object. (Inherited from Object.)
	Multiply(Matrix)	Multiplies the decomposition to a matrix. (Inherited from Decomposition.)
	Multiply(Vector)	Applies the base matrix of the decomposition to a vector. (Inherited from Decomposition.)
	MultiplyTranspose(Matrix)	Multiplies the transpose of the decomposition to a matrix. (Inherited from Decomposition.)
	MultiplyTranspose(Vector)	Multiplies the transpose of the decomposition to a vector. (Inherited from Decomposition.)
	Rank	Returns the numerical rank of a matrix. (Inherited from LinearOperator.)
	Rank(Double)	Returns the numerical rank of a matrix using the specified tolerance. (Overrides LinearOperatorRank(Double).)
	SetSingular	Sets a flag that indicates the underlying matrix of this decomposition is singular. (Inherited from Decomposition.)
	Solve(Matrix)	Solves the system of linear equations for the specified right-hand side matrix. (Inherited from LinearOperator.)
	Solve(Vector)	Solves the system of linear equations for the specified right-hand side vector. (Inherited from LinearOperator.)
	Solve(DenseMatrix, 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 LinearOperatorSolve(DenseMatrix, Boolean).)
	Solve(DenseVector, Boolean)	Solves the system of simultaneous linear equations for the specified right-hand side vector. (Overrides LinearOperatorSolve(DenseVector, Boolean).)
	SolveTo(DenseVector, DenseVector)	Solves the system for the specified right-hand side. (Inherited from LinearOperator.)
	SolveTo(Vector, DenseVector)	Solves the system for the specified right-hand side. (Inherited from LinearOperator.)
	SolveTranspose(Matrix)	Solves the transposed system of linear equations for the specified right-hand side matrix. (Inherited from LinearOperator.)
	SolveTranspose(Vector)	Solves the system of linear equations for the specified right-hand side vector. (Inherited from LinearOperator.)
	SolveTranspose(DenseMatrix, 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 LinearOperatorSolveTranspose(DenseMatrix, Boolean).)
	SolveTranspose(DenseVector, 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 LinearOperatorSolveTranspose(DenseVector, Boolean).)
	SolveTransposeTo	Solves the transposed system for the specified right-hand side. (Inherited from LinearOperator.)
	ToString	Returns a string that represents the current object. (Inherited from Object.)

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 of a real symmetric matrix are always real, 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(DenseMatrix, Boolean), GetInverse(Boolean) and EstimateConditionNumber methods simply call the corresponding method on the base matrix.

Numerical Libraries

Reference