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GeneralizedEigenvalueDecomposition Class |
Determines the Generalized eigenvalues and eigenvectors of two real square matrices.
Inheritance HierarchySystemObject
Accord.Math.DecompositionsGeneralizedEigenvalueDecomposition
Accord.Math.DecompositionsGeneralizedEigenvalueDecomposition
Namespace: Accord.Math.Decompositions
Assembly: Accord.Math (in Accord.Math.dll) Version: 3.8.0
SyntaxThe GeneralizedEigenvalueDecomposition type exposes the following members.
Constructors| Name | Description | |
|---|---|---|
 | GeneralizedEigenvalueDecomposition |
Constructs a new generalized eigenvalue decomposition.
|
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.
|
| RealAlphas | Returns the real parts of the alpha values. |
| RealEigenvalues | Returns the real parts of the eigenvalues. |
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.) |
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.) |
RemarksA 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 =
{
{ 1, 2, 3},
{ 8, 1, 4},
{ 3, 2, 3}
};
double[,] B =
{
{ 5, 1, 1},
{ 1, 5, 1},
{ 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 GeneralizedEigenvalueDecomposition(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
}
See Also