Represents the generalized eigenvalue decomposition of
two matrices.
SystemObject Extreme.Mathematics.LinearAlgebraGeneralizedDecompositionT Extreme.Mathematics.LinearAlgebraGeneralizedEigenvalueDecompositionT
Namespace:
Extreme.Mathematics.LinearAlgebra
Assembly:
Extreme.Numerics (in Extreme.Numerics.dll) Version: 8.1.1
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
Type Parameters
- T
The GeneralizedEigenvalueDecompositionT type exposes the following members.
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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.
GeneralizedEigenvalueDecompositionT
inherits fromGeneralizedDecompositionT.
Reference