RealEigenvalueDecomposition<T> Class

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

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

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

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

Inheritance

Object  →  LinearOperator<T>  →  Decomposition<T>  →  EigenvalueDecomposition<T>  →  RealEigenvalueDecomposition<T>

Type Parameters

T

Use the RealEigenvalueDecomposition<T> class to get access to properties specific to the eigenvalue decomposition of a real, square matrix.

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 and Eigenvectors properties return only the real eigenvalues and eigenvectors. The ComplexEigenvalues and ComplexEigenvectors properties return the full set of eigenvalues and eigenvectors. The RawEigenvectors property returns the real and imaginary parts of all eigenvectors.

The eigenvalues of a real symmetric matrix are always real, and its eigenvectors are real and orthogonal. Its eigenvalue decomposition can be calculated more easily, and there is no need to allow for complex eigenvectors and eigenvalues.

EigenvalueDecomposition<T> inherits from LinearOperator<T>. 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 eigen decomposition itself requires the solution of a system of equations. For this reason, the Solve(Vector<T>), GetInverse(Boolean) and EstimateConditionNumber() methods simply call the corresponding method on the base matrix.

Reference