Represents the eigenvalue decomposition of a
dense matrix.
SystemObject Extreme.Mathematics.LinearAlgebra.ComplexComplexLinearOperator Extreme.Mathematics.LinearAlgebra.ComplexComplexDecomposition Extreme.Mathematics.LinearAlgebraComplexEigenvalueDecomposition Extreme.Mathematics.LinearAlgebraHermitianEigenvalueDecomposition Extreme.Mathematics.LinearAlgebraNonHermitianEigenvalueDecomposition
Namespace: Extreme.Mathematics.LinearAlgebraAssembly: Extreme.Numerics.Net40 (in Extreme.Numerics.Net40.dll) Version: 6.0.16073.0 (6.0.17114.0)
public abstract class ComplexEigenvalueDecomposition : ComplexDecomposition
Public MustInherit Class ComplexEigenvalueDecomposition
Inherits ComplexDecomposition
public ref class ComplexEigenvalueDecomposition abstract : public ComplexDecomposition
[<AbstractClassAttribute>]
type ComplexEigenvalueDecomposition =
class
inherit ComplexDecomposition
end
The ComplexEigenvalueDecomposition type exposes the following members.
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The eigenvalue decomposition of a complex 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 complex or come in complex conjugate pairs.
The eigenvectors corresponding to complex eigenvalues are also complex. The eigenvectors
corresponding to pairs of complex conjugate eigenvalues are themselves complex
conjugates.
The eigenvalues of a complex symmetric matrix are always complex,
and its eigenvectors are orthogonal.
Its eigenvalue decomposition can be calculated more easily.
This is implemented by the SymmetricEigenvalueDecomposition class.
EigenvalueDecomposition inherits from
LinearOperator. However, unlike the other decompositions,
the nonsymmetric 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(ComplexDenseMatrix, Boolean), GetInverse(Boolean) and EstimateConditionNumber
methods simply call the corresponding method on the base matrix.
Numerical Libraries
Supported in: 5.x
Reference