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.

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.

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.

Deprecated:

since Eigen 3.3

Default constructor.

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

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()

Constructor; computes eigendecomposition of given matrix.

Parameters

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

This constructor calls compute() to compute the eigendecomposition.

Computes eigendecomposition of given matrix.

Parameters

[in]	matrix	Square matrix whose eigendecomposition is to be computed.
[in]	computeEigenvectors	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 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;

Output:

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

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

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

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

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;

Output:

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

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(0) << endl;

Output:

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

Reports whether previous computation was successful.

Returns

Success if computation was successful, NoConvergence otherwise.