JaggedGeneralizedEigenvalueDecomposition Class

Determines the Generalized eigenvalues and eigenvectors of two real square matrices.

Inheritance Hierarchy

SystemObject
Accord.Math.DecompositionsJaggedGeneralizedEigenvalueDecomposition

Namespace: Accord.Math.Decompositions
Assembly: Accord.Math (in Accord.Math.dll) Version: 3.8.0

Syntax

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public sealed class JaggedGeneralizedEigenvalueDecomposition : ICloneable

Public NotInheritable Class JaggedGeneralizedEigenvalueDecomposition
	Implements ICloneable

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The JaggedGeneralizedEigenvalueDecomposition type exposes the following members.

Constructors

	Name	Description
	JaggedGeneralizedEigenvalueDecomposition	Constructs a new generalized eigenvalue decomposition.

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Properties

	Name	Description
	Betas	Returns the beta values.
	DiagonalMatrix	Returns the block diagonal eigenvalue matrix.
	Eigenvectors	Returns the eigenvector matrix.
	ImaginaryAlphas	Returns the imaginary parts of the alpha values.
	ImaginaryEigenvalues	Returns the imaginary parts of the eigenvalues.
	IsDegenerate	Returns true if the eigenvalue problem is degenerate (ill-posed).
	IsSingular	Returns true if matrix B is singular.
	Rank	Returns the effective numerical matrix rank.
	RealAlphas	Returns the real parts of the alpha values.
	RealEigenvalues	Returns the real parts of the eigenvalues.

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Methods

	Name	Description
	Clone	Creates a new object that is a copy of the current instance.
	Equals	Determines whether the specified object is equal to the current object. (Inherited from Object.)
	GetHashCode	Serves as the default hash function. (Inherited from Object.)
	GetType	Gets the Type of the current instance. (Inherited from Object.)
	ToString	Returns a string that represents the current object. (Inherited from Object.)

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Extension Methods

	Name	Description
	HasMethod	Checks whether an object implements a method with the given name. (Defined by ExtensionMethods.)
	IsEqual	Compares two objects for equality, performing an elementwise comparison if the elements are vectors or matrices. (Defined by Matrix.)
	To(Type)	Overloaded. Converts an object into another type, irrespective of whether the conversion can be done at compile time or not. This can be used to convert generic types to numeric types during runtime. (Defined by ExtensionMethods.)
	ToT	Overloaded. Converts an object into another type, irrespective of whether the conversion can be done at compile time or not. This can be used to convert generic types to numeric types during runtime. (Defined by ExtensionMethods.)

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Remarks

A generalized eigenvalue problem is the problem of finding a vector v that obeys A * v = λ * B * v where A and B are matrices. If v obeys this equation, with some λ, then we call v the generalized eigenvector of A and B, and λ is called the generalized eigenvalue of A and B which corresponds to the generalized eigenvector v. The possible values of λ, must obey the identity det(A - λ*B) = 0.

Part of this code has been adapted from the original EISPACK routines in Fortran.

References:

Examples

// Suppose we have the following // matrices A and B shown below: double[][] A = { new double[] { 1, 2, 3}, new double[] { 8, 1, 4}, new double[] { 3, 2, 3} }; double[][] B = { new double[] { 5, 1, 1}, new double[] { 1, 5, 1}, new double[] { 1, 1, 5} }; // Now, suppose we would like to find values for λ // that are solutions for the equation det(A - λB) = 0 // For this, we can use a Generalized Eigendecomposition var gevd = new JaggedGeneralizedEigenvalueDecomposition(A, B); // Now, if A and B are Hermitian and B is positive // -definite, then the eigenvalues λ will be real: double[] lambda = gevd.RealEigenvalues; // Lets check if they are indeed a solution: for (int i = 0; i < lambda.Length; i++) { // Compute the determinant equation show above double det = Matrix.Determinant(A.Subtract(lambda[i].Multiply(B))); // almost zero }

Reference

Accord.Math.Decompositions Namespace