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.

	Name	Description
	ComplexEigenvalueDecomposition	Constructs a new EigenvalueDecomposition object.

	Name	Description
	BaseMatrix	Gets the underlying matrix of the decomposition. (Inherited from ComplexDecomposition.)
	Done	Gets or sets whether the decomposition has been performed. (Inherited from ComplexDecomposition.)
	Eigenvalues	Gets the eigenvalues of the matrix.
	Eigenvectors	Gets the eigenvectors of the matrix as a complex matrix.
	Factors	Gets a collection of the matrix factors that make up the decomposition. (Overrides ComplexDecompositionFactors.)
	Overwrite	Gets or sets a value indicating whether the BaseMatrix should be overwritten by its decomposition. (Inherited from ComplexDecomposition.)

	Name	Description
	Decompose	Performs the actual decomposition. (Inherited from ComplexDecomposition.)
	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 ComplexLinearOperatorEstimateConditionNumber.)
	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 ComplexLinearOperatorGetDeterminant.)
	GetHashCode	Serves as a hash function for a particular type. (Inherited from Object.)
	GetInverse	Calculates the inverse matrix.. (Inherited from ComplexLinearOperator.)
	GetInverse(Boolean)	Calculates the inverse of the factorized matrix. (Overrides ComplexLinearOperatorGetInverse(Boolean).)
	GetObjectData	Populates a SerializationInfo with the data needed to serialize the target object. (Inherited from ComplexLinearOperator.)
	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 ComplexDecomposition.)
	MemberwiseClone	Creates a shallow copy of the current Object. (Inherited from Object.)
	Multiply(ComplexMatrix)	Multiplies the decomposition to a matrix. (Inherited from ComplexDecomposition.)
	Multiply(ComplexVector)	Applies the base matrix of the decomposition to a vector. (Inherited from ComplexDecomposition.)
	MultiplyTranspose(ComplexMatrix)	Multiplies the transpose of the decomposition to a matrix. (Inherited from ComplexDecomposition.)
	MultiplyTranspose(ComplexVector)	Multiplies the transpose of the decomposition to a vector. (Inherited from ComplexDecomposition.)
	Rank	Returns the numerical rank of a matrix. (Inherited from ComplexLinearOperator.)
	Rank(Double)	Returns the numerical rank of a matrix using the specified tolerance. (Overrides ComplexLinearOperatorRank(Double).)
	SetSingular	Sets a flag that indicates the underlying matrix of this decomposition is singular. (Inherited from ComplexDecomposition.)
	Solve(ComplexMatrix)	Solves the system of linear equations for the specified right-hand side ComplexMatrix. (Inherited from ComplexLinearOperator.)
	Solve(ComplexVector)	Solves the system of linear equations for the specified right-hand side complex vector. (Inherited from ComplexLinearOperator.)
	Solve(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).)
	Solve(ComplexDenseVector, Boolean)	Solves the system of simultaneous linear equations for the specified right-hand side vector. (Overrides ComplexLinearOperatorSolve(ComplexDenseVector, Boolean).)
	SolveTranspose(ComplexMatrix)	Solves the transposed system of linear equations for the specified right-hand side ComplexMatrix. (Inherited from ComplexLinearOperator.)
	SolveTranspose(ComplexVector)	Solves the system of linear equations for the specified right-hand side complex vector. (Inherited from ComplexLinearOperator.)
	SolveTranspose(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).)
	SolveTranspose(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).)
	ToString	Returns a string that represents the current object. (Inherited from Object.)

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.

Numerical Libraries

Reference