Deprecated:

since Eigen 3.3

Real scalar type for _MatrixType.

This is just Scalar if Scalar is real (e.g., float or double), and the type of the real part of Scalar if Scalar is complex.

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

This is a column vector with entries of type RealScalar. The length of the vector is the size of _MatrixType.

Default constructor for fixed-size matrices.

The default constructor is useful in cases in which the user intends to perform decompositions via compute(). This constructor can only be used if _MatrixType is a fixed-size matrix; use SelfAdjointEigenSolver(Index) for dynamic-size matrices.

Example:

SelfAdjointEigenSolver<Matrix4f> es;
Matrix4f X = Matrix4f::Random(4,4);
Matrix4f A = X + X.transpose();
es.compute(A);
cout << "The eigenvalues of A are: " << es.eigenvalues().transpose() << endl;
es.compute(A + Matrix4f::Identity(4,4)); // 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:  -1.58 -0.473   1.32   2.46
The eigenvalues of A+I are: -0.581  0.527   2.32   3.46

Constructor, pre-allocates memory for dynamic-size matrices.

Parameters

[in]

size

Positive integer, size of the matrix whose eigenvalues and eigenvectors will be computed.

This constructor is useful for dynamic-size matrices, when the user intends to perform decompositions via compute(). The size parameter is only used as a hint. It is not an error to give a wrong size, but it may impair performance.

See also

compute() for an example

Constructor; computes eigendecomposition of given matrix.

Parameters

[in]	matrix	Selfadjoint matrix whose eigendecomposition is to be computed. Only the lower triangular part of the matrix is referenced.
[in]	options	Can be ComputeEigenvectors (default) or EigenvaluesOnly.

This constructor calls compute(const MatrixType&, int) to compute the eigenvalues of the matrix matrix. The eigenvectors are computed if options equals ComputeEigenvectors.

Example:

MatrixXd X = MatrixXd::Random(5,5);
MatrixXd A = X + X.transpose();
cout << "Here is a random symmetric 5x5 matrix, A:" << endl << A << endl << endl;
 
SelfAdjointEigenSolver<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;
 
double lambda = es.eigenvalues()[0];
cout << "Consider the first eigenvalue, lambda = " << lambda << endl;
VectorXd 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 * v << endl << endl;
 
MatrixXd D = es.eigenvalues().asDiagonal();
MatrixXd V = es.eigenvectors();
cout << "Finally, V * D * V^(-1) = " << endl << V * D * V.inverse() << endl;

Output:

Here is a random symmetric 5x5 matrix, A:
  1.36 -0.816  0.521   1.43 -0.144
-0.816 -0.659  0.794 -0.173 -0.406
 0.521  0.794 -0.541  0.461  0.179
  1.43 -0.173  0.461  -1.43  0.822
-0.144 -0.406  0.179  0.822  -1.37

The eigenvalues of A are:
 -2.65
 -1.77
-0.745
 0.227
  2.29
The matrix of eigenvectors, V, is:
 -0.326 -0.0984   0.347 -0.0109   0.874
 -0.207  -0.642   0.228   0.662  -0.232
 0.0495   0.629  -0.164    0.74   0.164
  0.721  -0.397  -0.402   0.115   0.385
 -0.573  -0.156  -0.799 -0.0256  0.0858

Consider the first eigenvalue, lambda = -2.65
If v is the corresponding eigenvector, then lambda * v = 
 0.865
  0.55
-0.131
 -1.91
  1.52
... and A * v = 
 0.865
  0.55
-0.131
 -1.91
  1.52

Finally, V * D * V^(-1) = 
  1.36 -0.816  0.521   1.43 -0.144
-0.816 -0.659  0.794 -0.173 -0.406
 0.521  0.794 -0.541  0.461  0.179
  1.43 -0.173  0.461  -1.43  0.822
-0.144 -0.406  0.179  0.822  -1.37

See also

compute(const MatrixType&, int)

Computes eigendecomposition of given matrix.

Parameters

[in]	matrix	Selfadjoint matrix whose eigendecomposition is to be computed. Only the lower triangular part of the matrix is referenced.
[in]	options	Can be ComputeEigenvectors (default) or EigenvaluesOnly.

Returns

Reference to *this

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

This implementation uses a symmetric QR algorithm. The matrix is first reduced to tridiagonal form using the Tridiagonalization class. The tridiagonal matrix is then brought to diagonal form with implicit symmetric QR steps with Wilkinson shift. Details can be found in Section 8.3 of Golub & Van Loan, Matrix Computations.

The cost of the computation is about \( 9n^3 \) if the eigenvectors are required and \( 4n^3/3 \) if they are not required.

This method reuses the memory in the SelfAdjointEigenSolver object that was allocated when the object was constructed, if the size of the matrix does not change.

Example:

SelfAdjointEigenSolver<MatrixXf> es(4);
MatrixXf X = MatrixXf::Random(4,4);
MatrixXf A = X + X.transpose();
es.compute(A);
cout << "The eigenvalues of A are: " << es.eigenvalues().transpose() << endl;
es.compute(A + MatrixXf::Identity(4,4)); // 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:  -1.58 -0.473   1.32   2.46
The eigenvalues of A+I are: -0.581  0.527   2.32   3.46

See also

SelfAdjointEigenSolver(const MatrixType&, int)

Computes eigendecomposition of given matrix using a closed-form algorithm.

This is a variant of compute(const MatrixType&, int options) which directly solves the underlying polynomial equation.

Currently only 2x2 and 3x3 matrices for which the sizes are known at compile time are supported (e.g., Matrix3d).

This method is usually significantly faster than the QR iterative algorithm but it might also be less accurate. It is also worth noting that for 3x3 matrices it involves trigonometric operations which are not necessarily available for all scalar types.

For the 3x3 case, we observed the following worst case relative error regarding the eigenvalues:

• double: 1e-8
• float: 1e-3

See also

compute(const MatrixType&, int options)

Computes the eigen decomposition from a tridiagonal symmetric matrix.

Parameters

[in]	diag	The vector containing the diagonal of the matrix.
[in]	subdiag	The subdiagonal of the matrix.
[in]	options	Can be ComputeEigenvectors (default) or EigenvaluesOnly.

Returns

Reference to *this

This function assumes that the matrix has been reduced to tridiagonal form.

See also

compute(const MatrixType&, int) for more information

Returns the eigenvalues of given matrix.

Returns

A const reference to the column vector containing the eigenvalues.

Precondition

The eigenvalues have been computed before.

The eigenvalues are repeated according to their algebraic multiplicity, so there are as many eigenvalues as rows in the matrix. The eigenvalues are sorted in increasing order.

Example:

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

Output:

The eigenvalues of the 3x3 matrix of ones are:
-3.09e-16
        0
        3

See also

eigenvectors(), MatrixBase::eigenvalues()

Returns the eigenvectors of given matrix.

Returns

A const reference to the matrix whose columns are the eigenvectors.

Precondition

The eigenvectors have been computed before.

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. If this object was used to solve the eigenproblem for the selfadjoint matrix \( A \), then the matrix returned by this function is the matrix \( V \) in the eigendecomposition \( A = V D V^{-1} \).

For a selfadjoint matrix, \( V \) is unitary, meaning its inverse is equal to its adjoint, \( V^{-1} = V^{\dagger} \). If \( A \) is real, then \( V \) is also real and therefore orthogonal, meaning its inverse is equal to its transpose, \( V^{-1} = V^T \).

Example:

MatrixXd ones = MatrixXd::Ones(3,3);
SelfAdjointEigenSolver<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
 0.408
 0.408

See also

eigenvalues()

Reports whether previous computation was successful.

Returns

Success if computation was successful, NoConvergence otherwise.

Computes the inverse square root of the matrix.

Returns

the inverse positive-definite square root of the matrix

Precondition

The eigenvalues and eigenvectors of a positive-definite matrix have been computed before.

This function uses the eigendecomposition \( A = V D V^{-1} \) to compute the inverse square root as \( V D^{-1/2} V^{-1} \). This is cheaper than first computing the square root with operatorSqrt() and then its inverse with MatrixBase::inverse().

Example:

MatrixXd X = MatrixXd::Random(4,4);
MatrixXd A = X * X.transpose();
cout << "Here is a random positive-definite matrix, A:" << endl << A << endl << endl;
 
SelfAdjointEigenSolver<MatrixXd> es(A);
cout << "The inverse square root of A is: " << endl;
cout << es.operatorInverseSqrt() << endl;
cout << "We can also compute it with operatorSqrt() and inverse(). That yields: " << endl;
cout << es.operatorSqrt().inverse() << endl;

Output:

Here is a random positive-definite matrix, A:
  1.41 -0.697 -0.111  0.508
-0.697  0.423 0.0991   -0.4
-0.111 0.0991   1.25  0.902
 0.508   -0.4  0.902    1.4

The inverse square root of A is: 
  1.88   2.78 -0.546  0.605
  2.78   8.61   -2.3   2.74
-0.546   -2.3   1.92  -1.36
 0.605   2.74  -1.36   2.18
We can also compute it with operatorSqrt() and inverse(). That yields: 
  1.88   2.78 -0.546  0.605
  2.78   8.61   -2.3   2.74
-0.546   -2.3   1.92  -1.36
 0.605   2.74  -1.36   2.18

See also

operatorSqrt(), MatrixBase::inverse(), MatrixFunctions Module

Computes the positive-definite square root of the matrix.

Returns

the positive-definite square root of the matrix

Precondition

The eigenvalues and eigenvectors of a positive-definite matrix have been computed before.

The square root of a positive-definite matrix \( A \) is the positive-definite matrix whose square equals \( A \). This function uses the eigendecomposition \( A = V D V^{-1} \) to compute the square root as \( A^{1/2} = V D^{1/2} V^{-1} \).

Example:

MatrixXd X = MatrixXd::Random(4,4);
MatrixXd A = X * X.transpose();
cout << "Here is a random positive-definite matrix, A:" << endl << A << endl << endl;
 
SelfAdjointEigenSolver<MatrixXd> es(A);
MatrixXd sqrtA = es.operatorSqrt();
cout << "The square root of A is: " << endl << sqrtA << endl;
cout << "If we square this, we get: " << endl << sqrtA*sqrtA << endl;

Output:

Here is a random positive-definite matrix, A:
  1.41 -0.697 -0.111  0.508
-0.697  0.423 0.0991   -0.4
-0.111 0.0991   1.25  0.902
 0.508   -0.4  0.902    1.4

The square root of A is: 
   1.09  -0.432 -0.0685     0.2
 -0.432   0.379   0.141  -0.269
-0.0685   0.141       1   0.468
    0.2  -0.269   0.468    1.04
If we square this, we get: 
  1.41 -0.697 -0.111  0.508
-0.697  0.423 0.0991   -0.4
-0.111 0.0991   1.25  0.902
 0.508   -0.4  0.902    1.4

See also

operatorInverseSqrt(), MatrixFunctions Module

Maximum number of iterations.

The algorithm terminates if it does not converge within m_maxIterations * n iterations, where n denotes the size of the matrix. This value is currently set to 30 (copied from LAPACK).