EigenvalueDecomposition<T> 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

Inheritance

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

Derived

Extreme.Mathematics.LinearAlgebra.NonHermitianEigenvalueDecomposition<T>

Extreme.Mathematics.LinearAlgebra.RealEigenvalueDecomposition<T>

Type Parameters

T

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.

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 eigenvalue 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