GeneralizedEigenvalueDecomposition<T> Class

public abstract class GeneralizedEigenvalueDecomposition<T> : GeneralizedDecomposition<T>

Public MustInherit Class GeneralizedEigenvalueDecomposition(Of T)
	Inherits GeneralizedDecomposition(Of T)

generic<typename T>
public ref class GeneralizedEigenvalueDecomposition abstract : public GeneralizedDecomposition<T>

[<AbstractClassAttribute>]
type GeneralizedEigenvalueDecomposition<'T> = 
    class
        inherit GeneralizedDecomposition<'T>
    end

Inheritance

Object  →  GeneralizedDecomposition<T>  →  GeneralizedEigenvalueDecomposition<T>

Type Parameters

T

The eigenvalue decomposition of a pair of square matrices A and B computes scalars ?,? and vectors x, y so that

Ax = ?Bx.

or

?Ay = By.

The scalars are called (generalized) eigenvalues and the vectors are called eigenvectors. If neither ? nor ? is zero, then the two problems are equivalent with x=y and ?=1/?. To cover the case where either may be zero or very small, two sets of values are returned, so that the eigenvalues are equal to their quotient.

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 generalized eigenvalues of a pair of real symmetric or complex Hermitian matrices where the secondary matrix is positive definite are always real,and its eigenvectors are orthogonal with respect to the second matrix. Its eigenvalue decomposition can be calculated more easily.

GeneralizedEigenvalueDecomposition<T> inherits fromGeneralizedDecomposition<T>.

Reference