EigenvalueDecompositionT Class

public abstract class EigenvalueDecomposition<T> : Decomposition<T>

Public MustInherit Class EigenvalueDecomposition(Of T)
	Inherits Decomposition(Of T)

generic<typename T>
public ref class EigenvalueDecomposition abstract : public Decomposition<T>

[<AbstractClassAttribute>]
type EigenvalueDecomposition<'T> =  
    class
        inherit Decomposition<'T>
    end

Type Parameters

T

The EigenvalueDecompositionT type exposes the following members.

	Name	Description
	EigenvalueDecompositionT	Constructs a new EigenvalueDecompositionT object.

	Name	Description
	BaseMatrix	Gets the underlying matrix of the decomposition. (Inherited from DecompositionT.)
	ColumnCount	Gets the number of columns in the matrix. (Inherited from LinearOperatorT.)
	ColumnIndex	Gets or sets the index of labels for the columns of the matrix. (Inherited from LinearOperatorT.)
	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 a value that indicates whether the decomposition has been performed. (Inherited from DecompositionT.)
	Eigenvalues	Gets the eigenvalues of the matrix.
	Eigenvectors	Gets the eigenvectors of the matrix.
	ElementType	Gets the element type of the matrix. (Inherited from LinearOperatorT.)
	Factors	Gets a collection of the matrix factors that make up the decomposition. (Overrides DecompositionTFactors.)
	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 DecompositionT.)
	RawEigenvectors	Gets the eigenvectors of the matrix.
	RowCount	Gets the number of rows in the matrix. (Inherited from LinearOperatorT.)
	RowIndex	Gets or sets the index of labels for the rows of the matrix. (Inherited from LinearOperatorT.)

	Name	Description
	AddProductToVector	Obsolete. Multiplies a VectorT by this instance and stores the result in a second vector of type DenseVectorT. (Inherited from LinearOperatorT.)
	Decompose	Performs the actual decomposition. (Inherited from DecompositionT.)
	Equals	Determines whether the specified object is equal to the current object. (Inherited from Object.)
	EstimateConditionNumber	Calculates an estimate for the condition number of a matrix. (Inherited from LinearOperatorT.)
	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 LinearOperatorTGetDeterminant.)
	GetHashCode	Serves as the default hash function. (Inherited from Object.)
	GetInverse	Calculates the inverse matrix.. (Inherited from LinearOperatorT.)
	GetInverse(Boolean)	Calculates the inverse of the factorized matrix. (Overrides LinearOperatorTGetInverse(Boolean).)
	GetObjectData	Populates a SerializationInfo with the data needed to serialize the target object. (Inherited from LinearOperatorT.)
	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 DecompositionT.)
	LeastSquaresSolve(MatrixT)	Solves the system of linear equations for the specified right-hand side matrix in the least squares sense. (Inherited from LinearOperatorT.)
	LeastSquaresSolve(VectorT)	Solves the system of linear equations for the specified right-hand side vector. (Inherited from LinearOperatorT.)
	LeastSquaresSolve(DenseMatrixT, Boolean)	Solves the system of linear equations for the specified right-hand side dense matrix in the least squares sense and optionally overwrites the right-hand side with the solution. (Inherited from LinearOperatorT.)
	LeastSquaresSolve(DenseVectorT, Boolean)	Solves the system of linear equations for the specified right-hand side dense vector in the least squares sense and optionally overwrites the right-hand side with the solution. (Inherited from LinearOperatorT.)
	LeastSquaresSolveInto(MatrixT, MatrixT)	Solves a system of equations defined by the matrix and multiple right-hand sides in the least squares sense. (Inherited from LinearOperatorT.)
	LeastSquaresSolveInto(VectorT, VectorT)	Solves a system of equations defined by the matrix and the specified right-hand side in the least squares sense. (Inherited from DecompositionT.)
	MemberwiseClone	Creates a shallow copy of the current Object. (Inherited from Object.)
	Multiply(MatrixT)	Applies the linear operator to a matrix. (Inherited from LinearOperatorT.)
	Multiply(VectorT)	Applies the linear operator to a vector. (Inherited from LinearOperatorT.)
	Multiply(TransposeOperation, MatrixT)	Obsolete. Multiplies a matrix on the left by this matrix. (Inherited from LinearOperatorT.)
	Multiply(TransposeOperation, VectorT)	Obsolete. Multiplies a vector on the left by this vector. (Inherited from LinearOperatorT.)
	MultiplyAndAddAsLeftFactorCore	Multiplies the matrix with a vector, adds the scaled result to another scaled vector, and returns the result. (Inherited from DecompositionT.)
	MultiplyAsLeftCore(TransposeOperation, MatrixT, MatrixT)	Multiplies a matrix by the linear operator and returns the result. (Inherited from DecompositionT.)
	MultiplyAsLeftCore(TransposeOperation, VectorT, VectorT)	Multiplies a vector by the linear operator and returns the result. (Inherited from DecompositionT.)
	MultiplyTranspose(MatrixT)	Applies the transpose of the linear operator to a matrix. (Inherited from LinearOperatorT.)
	MultiplyTranspose(VectorT)	Applies the (conjugate) transpose of the linear operator to a vector. (Inherited from LinearOperatorT.)
	Rank	Returns the numerical rank of a matrix. (Inherited from LinearOperatorT.)
	Rank(T)	Returns the numerical rank of a matrix using the specified tolerance. (Overrides LinearOperatorTRank(T).)
	SetSingular	Sets a flag that indicates the underlying matrix of this decomposition is singular. (Inherited from DecompositionT.)
	Solve(MatrixT)	Solves the system of linear equations for the specified right-hand side matrix. (Inherited from LinearOperatorT.)
	Solve(VectorT)	Solves the system of linear equations for the specified right-hand side vector. (Inherited from LinearOperatorT.)
	Solve(DenseMatrixT, Boolean)	Solves the system of linear equations for the specified right-hand side dense matrix and optionally overwrites the right-hand side with the solution. (Inherited from LinearOperatorT.)
	Solve(DenseVectorT, Boolean)	Solves the system of linear equations for the specified right-hand side dense vector and optionally overwrites the right-hand side with the solution. (Inherited from LinearOperatorT.)
	SolveInto(MatrixT, MatrixT)	Solves a system of equations defined by the matrix and multiple right-hand sides. (Inherited from LinearOperatorT.)
	SolveInto(VectorT, VectorT)	Solves a system of equations defined by the matrix and the specified right-hand side. (Inherited from LinearOperatorT.)
	SolveInto(TransposeOperation, MatrixT, MatrixT)	Solves a system of equations defined by the matrix and multiple right-hand sides. (Overrides LinearOperatorTSolveInto(TransposeOperation, MatrixT, MatrixT).)
	SolveInto(TransposeOperation, VectorT, VectorT)	Solves a system of equations defined by the matrix and the specified right-hand side. (Overrides LinearOperatorTSolveInto(TransposeOperation, VectorT, VectorT).)
	SolveTranspose(MatrixT)	Solves the transposed system of linear equations for the specified right-hand side matrix. (Inherited from LinearOperatorT.)
	SolveTranspose(VectorT)	Solves the system of linear equations for the specified right-hand side vector. (Inherited from LinearOperatorT.)
	SolveTranspose(DenseMatrixT, 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. (Inherited from LinearOperatorT.)
	SolveTranspose(DenseVectorT, 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. (Inherited from LinearOperatorT.)
	SolveTransposeInto(MatrixT, MatrixT)	Solves a system of equations defined by the matrix and multiple right-hand sides. (Inherited from LinearOperatorT.)
	SolveTransposeInto(VectorT, VectorT)	Solves a system of equations defined by the matrix and the specified right-hand side. (Inherited from LinearOperatorT.)
	ToString	Returns a string that represents the current object. (Inherited from Object.)

	Name	Description
	EstimateConditionNumberT	Calculates an estimate for the condition number of the matrix. (Defined by MatrixExtensions.)

The eigenvalue decomposition of a real square 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. Likewise, the eigenvalues of a complex Hermitian matrix are also real.

EigenvalueDecompositionT inherits from LinearOperatorT. However, unlike the other decompositions, the non-symmetric eigenvalue decomposition does not offer a speed advantage when performing calculations most calculations. For example, solving an equation using the eigenvalue decomposition itself requires the solution of a system of equations. For this reason, the Solve(VectorT), GetInverse(Boolean) and EstimateConditionNumber methods simply call the corresponding method on the base matrix.

Reference