Class ComplexEigenDecomposition
- Direct Known Subclasses:
OrderedComplexEigenDecomposition
Complex Eigen Decomposition differs from the EigenDecompositionSymmetric
since it
computes the eigen vectors as complex eigen vectors (if applicable).
Beware that in the complex case, you do not always have \(V \times V^{T} = I\) or even a diagonal matrix, even if the eigenvectors that form the columns of the V matrix are independent. On example is the square matrix \[ A = \left(\begin{matrix} 3 & -2\\ 4 & -1 \end{matrix}\right) \] which has two conjugate eigenvalues \(\lambda_1=1+2i\) and \(\lambda_2=1-2i\) with associated eigenvectors \(v_1^T = (1, 1-i)\) and \(v_2^T = (1, 1+i)\). \[ V\timesV^T = \left(\begin{matrix} 2 & 2\\ 2 & 0 \end{matrix}\right) \] which is not the identity matrix. Therefore, despite \(A \times V = V \times D\), \(A \ne V \times D \time V^T\), which would hold for real eigendecomposition.
Also note that for consistency with Wolfram langage eigenvectors, we add zero vectors when the geometric multiplicity of the eigenvalue is smaller than its algebraic multiplicity (hence the regular eigenvector matrix should be non-square). With these additional null vectors, the eigenvectors matrix becomes square. This happens for example with the square matrix \[ A = \left(\begin{matrix} 1 & 0 & 0\\ -2 & 1 & 0\\ 0 & 0 & 1 \end{matrix}\right) \] Its characteristic polynomial is \((1-\lambda)^3\), hence is has one eigen value \(\lambda=1\) with algebraic multiplicity 3. However, this eigenvalue leads to only two eigenvectors \(v_1=(0, 1, 0)\) and \(v_2=(0, 0, 1)\), hence its geometric multiplicity is only 2, not 3. So we add a third zero vector \(v_3=(0, 0, 0)\), in the same way Wolfram language does.
Compute complex eigen values from the Schur transform. Compute complex eigen vectors based on eigen values and the inverse iteration method. see: Inverse iteration Shifted inverse iteration Computation of matrix eigenvalues and eigenvectors-
Field Summary
Modifier and TypeFieldDescriptionstatic final double
Default threshold below which eigenvectors are considered equal.static final double
Default value to use for internal epsilon.static final double
Internally used epsilon criteria for final AV=VD check. -
Constructor Summary
ConstructorDescriptionComplexEigenDecomposition
(RealMatrix matrix) Constructor for decomposition.ComplexEigenDecomposition
(RealMatrix matrix, double eigenVectorsEquality, double epsilon, double epsilonAVVDCheck) Constructor for decomposition. -
Method Summary
Modifier and TypeMethodDescriptionprotected void
checkDefinition
(RealMatrix matrix) Check definition of the decomposition in runtime.protected void
findEigenValues
(RealMatrix matrix) Compute eigen values using the Schur transform.protected void
findEigenVectors
(FieldMatrix<Complex> matrix) Compute the eigen vectors using the inverse power method.getD()
Getter D.double
Computes the determinant.Complex[]
Getter of the eigen values.getEigenvector
(int i) Getter of the eigen vectors.getV()
Getter V.getVT()
Getter VT.boolean
Confirm if there are complex eigen values.protected void
Reset eigenvalues and eigen vectors from matrices.
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Field Details
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DEFAULT_EIGENVECTORS_EQUALITY
public static final double DEFAULT_EIGENVECTORS_EQUALITYDefault threshold below which eigenvectors are considered equal.- See Also:
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DEFAULT_EPSILON
public static final double DEFAULT_EPSILONDefault value to use for internal epsilon.- See Also:
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DEFAULT_EPSILON_AV_VD_CHECK
public static final double DEFAULT_EPSILON_AV_VD_CHECKInternally used epsilon criteria for final AV=VD check.- See Also:
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Constructor Details
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ComplexEigenDecomposition
Constructor for decomposition.This constructor uses the default values
DEFAULT_EIGENVECTORS_EQUALITY
,DEFAULT_EPSILON
andDEFAULT_EPSILON_AV_VD_CHECK
- Parameters:
matrix
- real matrix.
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ComplexEigenDecomposition
public ComplexEigenDecomposition(RealMatrix matrix, double eigenVectorsEquality, double epsilon, double epsilonAVVDCheck) Constructor for decomposition.The
eigenVectorsEquality
threshold is used to ensure the L∞-normalized eigenvectors found using inverse iteration are different from each other. if \(min(|e_i-e_j|,|e_i+e_j|)\) is smaller than this threshold, the algorithm considers it has found again an already known vector, so it drops it and attempts a new inverse iteration with a different start vector. This value should be much larger thanepsilon
which is used for convergence- Parameters:
matrix
- real matrix.eigenVectorsEquality
- threshold below which eigenvectors are considered equalepsilon
- Epsilon used for internal tests (e.g. is singular, eigenvalue ratio, etc.)epsilonAVVDCheck
- Epsilon criteria for final AV=VD check- Since:
- 1.8
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Method Details
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getEigenvalues
Getter of the eigen values.- Returns:
- eigen values.
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getEigenvector
Getter of the eigen vectors.- Parameters:
i
- which eigen vector.- Returns:
- eigen vector.
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matricesToEigenArrays
protected void matricesToEigenArrays()Reset eigenvalues and eigen vectors from matrices.This method is intended to be called by sub-classes (mainly
OrderedComplexEigenDecomposition
) that reorder the matrices elements. It rebuild the eigenvalues and eigen vectors arrays from the D and V matrices.- Since:
- 2.1
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hasComplexEigenvalues
public boolean hasComplexEigenvalues()Confirm if there are complex eigen values.- Returns:
- true if there are complex eigen values.
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getDeterminant
public double getDeterminant()Computes the determinant.- Returns:
- the determinant.
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getV
Getter V.- Returns:
- V.
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getD
Getter D.- Returns:
- D.
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getVT
Getter VT.- Returns:
- VT.
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findEigenValues
Compute eigen values using the Schur transform.- Parameters:
matrix
- real matrix to compute eigen values.
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findEigenVectors
Compute the eigen vectors using the inverse power method.- Parameters:
matrix
- real matrix to compute eigen vectors.
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checkDefinition
Check definition of the decomposition in runtime.- Parameters:
matrix
- matrix to be decomposed.
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