Public Types | Public Member Functions | List of all members
ComplexEigenSolver< _MatrixType > Class Template Reference

Computes eigenvalues and eigenvectors of general complex matrices. More...

#include <ComplexEigenSolver.h>

Public Types

typedef std::complex< RealScalar > ComplexScalar
 Complex scalar type for MatrixType.
 
typedef Matrix< ComplexScalar, ColsAtCompileTime, 1, Options &(~RowMajor), MaxColsAtCompileTime, 1 > EigenvalueType
 Type for vector of eigenvalues as returned by eigenvalues().
 
typedef Matrix< ComplexScalar, RowsAtCompileTime, ColsAtCompileTime, Options, MaxRowsAtCompileTime, MaxColsAtCompileTime > EigenvectorType
 Type for matrix of eigenvectors as returned by eigenvectors().
 
typedef _MatrixType MatrixType
 Synonym for the template parameter _MatrixType.
 
typedef MatrixType::Scalar Scalar
 Scalar type for matrices of type MatrixType.
 

Public Member Functions

 ComplexEigenSolver ()
 Default constructor.
 
 ComplexEigenSolver (const MatrixType &matrix, bool computeEigenvectors=true)
 Constructor; computes eigendecomposition of given matrix.
 
 ComplexEigenSolver (Index size)
 Default Constructor with memory preallocation.
 
ComplexEigenSolver & compute (const MatrixType &matrix, bool computeEigenvectors=true)
 Computes eigendecomposition of given matrix.
 
const EigenvalueType & eigenvalues () const
 Returns the eigenvalues of given matrix.
 
const EigenvectorType & eigenvectors () const
 Returns the eigenvectors of given matrix.
 
ComputationInfo info () const
 Reports whether previous computation was successful.
 

Detailed Description

template<typename _MatrixType>
class Eigen::ComplexEigenSolver< _MatrixType >

Computes eigenvalues and eigenvectors of general complex matrices.

This is defined in the Eigenvalues module.

#include <Eigen/Eigenvalues>
Template Parameters
_MatrixTypethe type of the matrix of which we are computing the eigendecomposition; this is expected to be an instantiation of the Matrix class template.

The eigenvalues and eigenvectors of a matrix $ A $ are scalars $ \lambda $ and vectors $ v $ such that $ Av = \lambda v $. If $ D $ is a diagonal matrix with the eigenvalues on the diagonal, and $ V $ is a matrix with the eigenvectors as its columns, then $ A V = V D $. The matrix $ V $ is almost always invertible, in which case we have $ A = V D V^{-1} $. This is called the eigendecomposition.

The main function in this class is compute(), which computes the eigenvalues and eigenvectors of a given function. The documentation for that function contains an example showing the main features of the class.

See also
class EigenSolver, class SelfAdjointEigenSolver

Member Typedef Documentation

◆ ComplexScalar

template<typename _MatrixType>
typedef std::complex<RealScalar> ComplexScalar

Complex scalar type for MatrixType.

This is std::complex<Scalar> if Scalar is real (e.g., float or double) and just Scalar if Scalar is complex.

◆ EigenvalueType

template<typename _MatrixType>
typedef Matrix<ComplexScalar, ColsAtCompileTime, 1, Options&(~RowMajor), MaxColsAtCompileTime, 1> EigenvalueType

Type for vector of eigenvalues as returned by eigenvalues().

This is a column vector with entries of type ComplexScalar. The length of the vector is the size of MatrixType.

◆ EigenvectorType

template<typename _MatrixType>
typedef Matrix<ComplexScalar, RowsAtCompileTime, ColsAtCompileTime, Options, MaxRowsAtCompileTime, MaxColsAtCompileTime> EigenvectorType

Type for matrix of eigenvectors as returned by eigenvectors().

This is a square matrix with entries of type ComplexScalar. The size is the same as the size of MatrixType.

Constructor & Destructor Documentation

◆ ComplexEigenSolver() [1/3]

template<typename _MatrixType>
ComplexEigenSolver ( )
inline

Default constructor.

The default constructor is useful in cases in which the user intends to perform decompositions via compute().

Referenced by compute().

◆ ComplexEigenSolver() [2/3]

template<typename _MatrixType>
ComplexEigenSolver ( Index size)
inline

Default Constructor with memory preallocation.

Like the default constructor but with preallocation of the internal data according to the specified problem size.

See also
ComplexEigenSolver()

◆ ComplexEigenSolver() [3/3]

template<typename _MatrixType>
ComplexEigenSolver ( const MatrixType & matrix,
bool computeEigenvectors = true )
inline

Constructor; computes eigendecomposition of given matrix.

Parameters
[in]matrixSquare matrix whose eigendecomposition is to be computed.
[in]computeEigenvectorsIf true, both the eigenvectors and the eigenvalues are computed; if false, only the eigenvalues are computed.

This constructor calls compute() to compute the eigendecomposition.

References compute().

Member Function Documentation

◆ compute()

template<typename MatrixType>
ComplexEigenSolver< MatrixType > & compute ( const MatrixType & matrix,
bool computeEigenvectors = true )

Computes eigendecomposition of given matrix.

Parameters
[in]matrixSquare matrix whose eigendecomposition is to be computed.
[in]computeEigenvectorsIf true, both the eigenvectors and the eigenvalues are computed; if false, only the eigenvalues are computed.
Returns
Reference to *this

This function computes the eigenvalues of the complex matrix matrix. The eigenvalues() function can be used to retrieve them. If computeEigenvectors is true, then the eigenvectors are also computed and can be retrieved by calling eigenvectors().

The matrix is first reduced to Schur form using the ComplexSchur class. The Schur decomposition is then used to compute the eigenvalues and eigenvectors.

The cost of the computation is dominated by the cost of the Schur decomposition, which is $ O(n^3) $ where $ n $ is the size of the matrix.

Example:

MatrixXcf A = MatrixXcf::Random(4,4);
cout << "Here is a random 4x4 matrix, A:" << endl << A << endl << endl;
ComplexEigenSolver<MatrixXcf> ces;
ces.compute(A);
cout << "The eigenvalues of A are:" << endl << ces.eigenvalues() << endl;
cout << "The matrix of eigenvectors, V, is:" << endl << ces.eigenvectors() << endl << endl;
complex<float> lambda = ces.eigenvalues()[0];
cout << "Consider the first eigenvalue, lambda = " << lambda << endl;
VectorXcf v = ces.eigenvectors().col(0);
cout << "If v is the corresponding eigenvector, then lambda * v = " << endl << lambda * v << endl;
cout << "... and A * v = " << endl << A * v << endl << endl;
cout << "Finally, V * D * V^(-1) = " << endl
<< ces.eigenvectors() * ces.eigenvalues().asDiagonal() * ces.eigenvectors().inverse() << endl;
Eigen::ComplexEigenSolver::compute
ComplexEigenSolver & compute(const MatrixType &matrix, bool computeEigenvectors=true)
Computes eigendecomposition of given matrix.
Definition ComplexEigenSolver.h:238
Eigen::ComplexEigenSolver::ComplexEigenSolver
ComplexEigenSolver()
Default constructor.
Definition ComplexEigenSolver.h:92
Eigen::ComplexEigenSolver::eigenvectors
const EigenvectorType & eigenvectors() const
Returns the eigenvectors of given matrix.
Definition ComplexEigenSolver.h:156
Eigen::ComplexEigenSolver::eigenvalues
const EigenvalueType & eigenvalues() const
Returns the eigenvalues of given matrix.
Definition ComplexEigenSolver.h:181
Eigen::Matrix< std::complex< float >, Dynamic, Dynamic >::Random
static const CwiseNullaryOp< internal::scalar_random_op< Scalar >, Matrix< std::complex< float >, _Rows, _Cols, _Options, _MaxRows, _MaxCols > > Random(Index rows, Index cols)
Definition Random.h:49
Eigen::MatrixBase< Matrix< _Scalar, _Rows, _Cols, _Options, _MaxRows, _MaxCols > >::asDiagonal
const DiagonalWrapper< const Matrix< _Scalar, _Rows, _Cols, _Options, _MaxRows, _MaxCols > > asDiagonal() const
Definition DiagonalMatrix.h:273
Eigen::MatrixBase< Matrix< _Scalar, _Rows, _Cols, _Options, _MaxRows, _MaxCols > >::inverse
const internal::inverse_impl< Matrix< _Scalar, _Rows, _Cols, _Options, _MaxRows, _MaxCols > > inverse() const
Definition Inverse.h:316

Output: 

Here is a random 4x4 matrix, A:
  (0.68,-0.211)  (-0.444,0.108)   (0.271,0.435) (-0.687,-0.198)
  (0.566,0.597) (-0.0452,0.258)  (-0.717,0.214)  (-0.74,-0.782)
 (0.823,-0.605)  (-0.27,0.0268) (-0.967,-0.514)  (0.998,-0.563)
  (-0.33,0.536)   (0.904,0.832)  (-0.726,0.608)  (0.0259,0.678)

The eigenvalues of A are:
(0.505,0.137)
(1.22,-0.758)
(-0.402,1.52)
(-1.63,-0.691)
The matrix of eigenvectors, V, is:
   (0.246,-0.106)     (0.263,0.418)   (0.296,-0.0417)   (-0.122,-0.271)
   (0.205,-0.629)    (-0.457,0.466)    (0.456,-0.244)     (0.247,-0.23)
  (0.432,-0.0359) (-0.0146,-0.0651)    (-0.334,0.191)    (0.859,0.0877)
     (0.301,0.46)    (-0.397,-0.41)   (-0.328,-0.623)   (-0.116,-0.195)

Consider the first eigenvalue, lambda = (0.505,0.137)
If v is the corresponding eigenvector, then lambda * v = 
(0.139,-0.0197)
(0.19,-0.29)
(0.223,0.0412)
(0.0891,0.274)
... and A * v = 
(0.139,-0.0197)
(0.19,-0.29)
(0.223,0.0412)
(0.0891,0.274)

Finally, V * D * V^(-1) = 
  (0.68,-0.211)  (-0.444,0.108)   (0.271,0.435) (-0.687,-0.198)
  (0.566,0.597) (-0.0452,0.258)  (-0.717,0.214)  (-0.74,-0.782)
 (0.823,-0.605)  (-0.27,0.0268) (-0.967,-0.514)  (0.998,-0.563)
  (-0.33,0.536)   (0.904,0.832)  (-0.726,0.608)  (0.0259,0.678)

References ComplexEigenSolver(), and Eigen::Success.

Referenced by ComplexEigenSolver().

◆ eigenvalues()

template<typename _MatrixType>
const EigenvalueType & eigenvalues ( ) const
inline

Returns the eigenvalues of given matrix.

Returns
A const reference to the column vector containing the eigenvalues.
Precondition
Either the constructor ComplexEigenSolver(const MatrixType& matrix, bool) or the member function compute(const MatrixType& matrix, bool) has been called before to compute the eigendecomposition of a matrix.

This function returns a column vector containing the eigenvalues. Eigenvalues are repeated according to their algebraic multiplicity, so there are as many eigenvalues as rows in the matrix. The eigenvalues are not sorted in any particular order.

Example:

MatrixXcf ones = MatrixXcf::Ones(3,3);
ComplexEigenSolver<MatrixXcf> ces(ones, /* computeEigenvectors = */ false);
cout << "The eigenvalues of the 3x3 matrix of ones are:"
<< endl << ces.eigenvalues() << endl;
Eigen::Matrix< std::complex< float >, Dynamic, Dynamic >::Ones
static const ConstantReturnType Ones(Index rows, Index cols)
Definition CwiseNullaryOp.h:557

Output: 

The eigenvalues of the 3x3 matrix of ones are:
(0,-0)
(0,0)
(3,0)

◆ eigenvectors()

template<typename _MatrixType>
const EigenvectorType & eigenvectors ( ) const
inline

Returns the eigenvectors of given matrix.

Returns
A const reference to the matrix whose columns are the eigenvectors.
Precondition
Either the constructor ComplexEigenSolver(const MatrixType& matrix, bool) or the member function compute(const MatrixType& matrix, bool) has been called before to compute the eigendecomposition of a matrix, and computeEigenvectors was set to true (the default).

This function returns a matrix whose columns are the eigenvectors. Column $ k $ is an eigenvector corresponding to eigenvalue number $ k $ as returned by eigenvalues(). The eigenvectors are normalized to have (Euclidean) norm equal to one. The matrix returned by this function is the matrix $ V $ in the eigendecomposition $ A = V D V^{-1} $, if it exists.

Example:

MatrixXcf ones = MatrixXcf::Ones(3,3);
ComplexEigenSolver<MatrixXcf> ces(ones);
cout << "The first eigenvector of the 3x3 matrix of ones is:"
<< endl << ces.eigenvectors().col(1) << endl;

Output: 

The first eigenvector of the 3x3 matrix of ones is:
(0.154,0)
(-0.772,0)
(0.617,0)

◆ info()

template<typename _MatrixType>
ComputationInfo info ( ) const
inline

Reports whether previous computation was successful.

Returns
Success if computation was succesful, NoConvergence otherwise.
The documentation for this class was generated from the following file: