template<typename _MatrixType>
Eigen::EigenSolver class

Computes eigenvalues and eigenvectors of general matrices.

Template parameters
_MatrixType the type of the matrix of which we are computing the eigendecomposition; this is expected to be an instantiation of the Matrix class template. Currently, only real matrices are supported.

Contents

This is defined in the Eigenvalues module. #include <Eigen/Eigenvalues>

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 eigenvalues and eigenvectors of a matrix may be complex, even when the matrix is real. However, we can choose real matrices $ V $ and $ D $ satisfying $ A V = V D $ , just like the eigendecomposition, if the matrix $ D $ is not required to be diagonal, but if it is allowed to have blocks of the form

\[ \begin{bmatrix} u & v \\ -v & u \end{bmatrix} \]

(where $ u $ and $ v $ are real numbers) on the diagonal. These blocks correspond to complex eigenvalue pairs $ u \pm iv $ . We call this variant of the eigendecomposition the pseudo-eigendecomposition.

Call the function compute() to compute the eigenvalues and eigenvectors of a given matrix. Alternatively, you can use the EigenSolver(const MatrixType&, bool) constructor which computes the eigenvalues and eigenvectors at construction time. Once the eigenvalue and eigenvectors are computed, they can be retrieved with the eigenvalues() and eigenvectors() functions. The pseudoEigenvalueMatrix() and pseudoEigenvectors() methods allow the construction of the pseudo-eigendecomposition.

The documentation for EigenSolver(const MatrixType&, bool) contains an example of the typical use of this class.

Public types

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

Constructors, destructors, conversion operators

EigenSolver()
Default constructor.
EigenSolver(Index size) explicit
Default constructor with memory preallocation.
template<typename InputType>
EigenSolver(const EigenBase<InputType>& matrix, bool computeEigenvectors = true) explicit
Constructor; computes eigendecomposition of given matrix.

Public functions

template<typename InputType>
auto compute(const EigenBase<InputType>& matrix, bool computeEigenvectors = true) -> EigenSolver&
Computes eigendecomposition of given matrix.
auto eigenvalues() const -> const EigenvalueType&
Returns the eigenvalues of given matrix.
auto eigenvectors() const -> EigenvectorsType
Returns the eigenvectors of given matrix.
auto getMaxIterations() -> Index
Returns the maximum number of iterations.
auto info() const -> ComputationInfo
auto pseudoEigenvalueMatrix() const -> MatrixType
Returns the block-diagonal matrix in the pseudo-eigendecomposition.
auto pseudoEigenvectors() const -> const MatrixType&
Returns the pseudo-eigenvectors of given matrix.
auto setMaxIterations(Index maxIters) -> EigenSolver&
Sets the maximum number of iterations allowed.

Typedef documentation

template<typename _MatrixType>
typedef std::complex<RealScalar> Eigen::EigenSolver<_MatrixType>::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.

template<typename _MatrixType>
typedef Matrix<ComplexScalar, ColsAtCompileTime, 1, Options&~RowMajor, MaxColsAtCompileTime, 1> Eigen::EigenSolver<_MatrixType>::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.

template<typename _MatrixType>
typedef Matrix<ComplexScalar, RowsAtCompileTime, ColsAtCompileTime, Options, MaxRowsAtCompileTime, MaxColsAtCompileTime> Eigen::EigenSolver<_MatrixType>::EigenvectorsType

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.

template<typename _MatrixType>
typedef Eigen::Index Eigen::EigenSolver<_MatrixType>::Index

Function documentation

template<typename _MatrixType>
Eigen::EigenSolver<_MatrixType>::EigenSolver()

Default constructor.

The default constructor is useful in cases in which the user intends to perform decompositions via EigenSolver::compute(const MatrixType&, bool).

template<typename _MatrixType>
Eigen::EigenSolver<_MatrixType>::EigenSolver(Index size) explicit

Default constructor with memory preallocation.

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

template<typename _MatrixType> template<typename InputType>
Eigen::EigenSolver<_MatrixType>::EigenSolver(const EigenBase<InputType>& matrix, bool computeEigenvectors = true) explicit

Constructor; computes eigendecomposition of given matrix.

Parameters
matrix in Square matrix whose eigendecomposition is to be computed.
computeEigenvectors in If true, both the eigenvectors and the eigenvalues are computed; if false, only the eigenvalues are computed.

This constructor calls compute() to compute the eigenvalues and eigenvectors.

Example:

MatrixXd A = MatrixXd::Random(6,6);
cout << "Here is a random 6x6 matrix, A:" << endl << A << endl << endl;

EigenSolver<MatrixXd> es(A);
cout << "The eigenvalues of A are:" << endl << es.eigenvalues() << endl;
cout << "The matrix of eigenvectors, V, is:" << endl << es.eigenvectors() << endl << endl;

complex<double> lambda = es.eigenvalues()[0];
cout << "Consider the first eigenvalue, lambda = " << lambda << endl;
VectorXcd v = es.eigenvectors().col(0);
cout << "If v is the corresponding eigenvector, then lambda * v = " << endl << lambda * v << endl;
cout << "... and A * v = " << endl << A.cast<complex<double> >() * v << endl << endl;

MatrixXcd D = es.eigenvalues().asDiagonal();
MatrixXcd V = es.eigenvectors();
cout << "Finally, V * D * V^(-1) = " << endl << V * D * V.inverse() << endl;

Output:

Here is a random 6x6 matrix, A:
   0.68   -0.33   -0.27  -0.717  -0.687  0.0259
 -0.211   0.536  0.0268   0.214  -0.198   0.678
  0.566  -0.444   0.904  -0.967   -0.74   0.225
  0.597   0.108   0.832  -0.514  -0.782  -0.408
  0.823 -0.0452   0.271  -0.726   0.998   0.275
 -0.605   0.258   0.435   0.608  -0.563  0.0486

The eigenvalues of A are:
  (0.049,1.06)
 (0.049,-1.06)
     (0.967,0)
     (0.353,0)
 (0.618,0.129)
(0.618,-0.129)
The matrix of eigenvectors, V, is:
 (-0.292,-0.454)   (-0.292,0.454)      (-0.0607,0)       (-0.733,0)    (0.59,-0.121)     (0.59,0.121)
  (0.134,-0.104)    (0.134,0.104)       (-0.799,0)        (0.136,0)    (0.334,0.368)   (0.334,-0.368)
  (-0.422,-0.18)    (-0.422,0.18)        (0.192,0)       (0.0563,0)  (-0.335,-0.143)   (-0.335,0.143)
 (-0.589,0.0274) (-0.589,-0.0274)      (-0.0788,0)       (-0.627,0)   (0.322,-0.155)    (0.322,0.155)
  (-0.248,0.132)  (-0.248,-0.132)        (0.401,0)        (0.218,0) (-0.335,-0.0761)  (-0.335,0.0761)
    (0.105,0.18)    (0.105,-0.18)       (-0.392,0)     (-0.00564,0)  (-0.0324,0.103) (-0.0324,-0.103)

Consider the first eigenvalue, lambda = (0.049,1.06)
If v is the corresponding eigenvector, then lambda * v = 
  (0.466,-0.331)
   (0.117,0.137)
   (0.17,-0.456)
(-0.0578,-0.622)
 (-0.152,-0.256)
   (-0.186,0.12)
... and A * v = 
  (0.466,-0.331)
   (0.117,0.137)
   (0.17,-0.456)
(-0.0578,-0.622)
 (-0.152,-0.256)
   (-0.186,0.12)

Finally, V * D * V^(-1) = 
   (0.68,-4.44e-16)   (-0.33,-5.55e-17)   (-0.27,-1.11e-16)  (-0.717,-4.44e-16)   (-0.687,8.88e-16)          (0.0259,0)
  (-0.211,2.22e-16)    (0.536,1.91e-17)          (0.0268,0)           (0.214,0)   (-0.198,1.33e-15)           (0.678,0)
   (0.566,2.22e-16)  (-0.444,-1.53e-16)   (0.904,-2.22e-16)  (-0.967,-1.11e-16)    (-0.74,4.44e-16)    (0.225,2.22e-16)
  (0.597,-2.22e-16)   (0.108,-2.78e-16)   (0.832,-2.22e-16)  (-0.514,-1.11e-16)          (-0.782,0)  (-0.408,-2.22e-16)
  (0.823,-2.22e-16) (-0.0452,-1.67e-16)           (0.271,0)   (-0.726,1.11e-16)   (0.998,-8.88e-16)    (0.275,4.44e-16)
  (-0.605,2.91e-16)   (0.258,-6.94e-18)   (0.435,-6.94e-17)    (0.608,1.39e-17)   (-0.563,5.27e-16)   (0.0486,7.11e-17)

template<typename _MatrixType> template<typename InputType>
EigenSolver& Eigen::EigenSolver<_MatrixType>::compute(const EigenBase<InputType>& matrix, bool computeEigenvectors = true)

Computes eigendecomposition of given matrix.

Parameters
matrix in Square matrix whose eigendecomposition is to be computed.
computeEigenvectors in If 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 real 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 real Schur form using the RealSchur 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 very approximately $ 25n^3 $ (where $ n $ is the size of the matrix) if computeEigenvectors is true, and $ 10n^3 $ if computeEigenvectors is false.

This method reuses of the allocated data in the EigenSolver object.

Example:

EigenSolver<MatrixXf> es;
MatrixXf A = MatrixXf::Random(4,4);
es.compute(A, /* computeEigenvectors = */ false);
cout << "The eigenvalues of A are: " << es.eigenvalues().transpose() << endl;
es.compute(A + MatrixXf::Identity(4,4), false); // re-use es to compute eigenvalues of A+I
cout << "The eigenvalues of A+I are: " << es.eigenvalues().transpose() << endl;

Output:

The eigenvalues of A are:    (0.755,0.528)   (0.755,-0.528)  (-0.323,0.0965) (-0.323,-0.0965)
The eigenvalues of A+I are:    (1.75,0.528)   (1.75,-0.528)  (0.677,0.0965) (0.677,-0.0965)

template<typename _MatrixType>
const EigenvalueType& Eigen::EigenSolver<_MatrixType>::eigenvalues() const

Returns the eigenvalues of given matrix.

Returns A const reference to the column vector containing the eigenvalues.

The 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:

MatrixXd ones = MatrixXd::Ones(3,3);
EigenSolver<MatrixXd> es(ones, false);
cout << "The eigenvalues of the 3x3 matrix of ones are:" 
     << endl << es.eigenvalues() << endl;

Output:

The eigenvalues of the 3x3 matrix of ones are:
(-5.31e-17,0)
        (3,0)
        (0,0)

template<typename _MatrixType>
EigenvectorsType Eigen::EigenSolver<_MatrixType>::eigenvectors() const

Returns the eigenvectors of given matrix.

Returns Matrix whose columns are the (possibly complex) eigenvectors.

Column $ k $ of the returned matrix 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:

MatrixXd ones = MatrixXd::Ones(3,3);
EigenSolver<MatrixXd> es(ones);
cout << "The first eigenvector of the 3x3 matrix of ones is:"
     << endl << es.eigenvectors().col(0) << endl;

Output:

The first eigenvector of the 3x3 matrix of ones is:
(-0.816,0)
 (0.408,0)
 (0.408,0)

template<typename _MatrixType>
ComputationInfo Eigen::EigenSolver<_MatrixType>::info() const

Returns NumericalIssue if the input contains INF or NaN values or overflow occurred. Returns Success otherwise.

template<typename _MatrixType>
MatrixType Eigen::EigenSolver<_MatrixType>::pseudoEigenvalueMatrix() const

Returns the block-diagonal matrix in the pseudo-eigendecomposition.

Returns A block-diagonal matrix.

The matrix $ D $ returned by this function is real and block-diagonal. The blocks on the diagonal are either 1-by-1 or 2-by-2 blocks of the form $ \begin{bmatrix} u & v \\ -v & u \end{bmatrix} $ . These blocks are not sorted in any particular order. The matrix $ D $ and the matrix $ V $ returned by pseudoEigenvectors() satisfy $ AV = VD $ .

template<typename _MatrixType>
const MatrixType& Eigen::EigenSolver<_MatrixType>::pseudoEigenvectors() const

Returns the pseudo-eigenvectors of given matrix.

Returns Const reference to matrix whose columns are the pseudo-eigenvectors.

The real matrix $ V $ returned by this function and the block-diagonal matrix $ D $ returned by pseudoEigenvalueMatrix() satisfy $ AV = VD $ .

Example:

MatrixXd A = MatrixXd::Random(6,6);
cout << "Here is a random 6x6 matrix, A:" << endl << A << endl << endl;

EigenSolver<MatrixXd> es(A);
MatrixXd D = es.pseudoEigenvalueMatrix();
MatrixXd V = es.pseudoEigenvectors();
cout << "The pseudo-eigenvalue matrix D is:" << endl << D << endl;
cout << "The pseudo-eigenvector matrix V is:" << endl << V << endl;
cout << "Finally, V * D * V^(-1) = " << endl << V * D * V.inverse() << endl;

Output:

Here is a random 6x6 matrix, A:
   0.68   -0.33   -0.27  -0.717  -0.687  0.0259
 -0.211   0.536  0.0268   0.214  -0.198   0.678
  0.566  -0.444   0.904  -0.967   -0.74   0.225
  0.597   0.108   0.832  -0.514  -0.782  -0.408
  0.823 -0.0452   0.271  -0.726   0.998   0.275
 -0.605   0.258   0.435   0.608  -0.563  0.0486

The pseudo-eigenvalue matrix D is:
 0.049   1.06      0      0      0      0
 -1.06  0.049      0      0      0      0
     0      0  0.967      0      0      0
     0      0      0  0.353      0      0
     0      0      0      0  0.618  0.129
     0      0      0      0 -0.129  0.618
The pseudo-eigenvector matrix V is:
  -0.571   -0.888   -0.066    -1.13     17.2    -3.53
   0.263   -0.204   -0.869     0.21     9.73     10.7
  -0.827   -0.352    0.209   0.0871    -9.74    -4.17
   -1.15   0.0535  -0.0857   -0.971     9.36    -4.52
  -0.485    0.258    0.436    0.337    -9.74    -2.21
   0.206    0.353   -0.426 -0.00873   -0.944     2.98
Finally, V * D * V^(-1) = 
   0.68   -0.33   -0.27  -0.717  -0.687  0.0259
 -0.211   0.536  0.0268   0.214  -0.198   0.678
  0.566  -0.444   0.904  -0.967   -0.74   0.225
  0.597   0.108   0.832  -0.514  -0.782  -0.408
  0.823 -0.0452   0.271  -0.726   0.998   0.275
 -0.605   0.258   0.435   0.608  -0.563  0.0486