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 GeneralizedSelfAdjointEigenSolver(Index) for dynamic-size matrices.

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 generalized eigendecomposition of given matrix pencil.

Parameters

[in]	matA	Selfadjoint matrix in matrix pencil. Only the lower triangular part of the matrix is referenced.
[in]	matB	Positive-definite matrix in matrix pencil. Only the lower triangular part of the matrix is referenced.
[in]	options	A or-ed set of flags {ComputeEigenvectors,EigenvaluesOnly} | {Ax_lBx,ABx_lx,BAx_lx}. Default is ComputeEigenvectors|Ax_lBx.

This constructor calls compute(const MatrixType&, const MatrixType&, int) to compute the eigenvalues and (if requested) the eigenvectors of the generalized eigenproblem \( Ax = \lambda B x \) with matA the selfadjoint matrix \( A \) and matB the positive definite matrix \( B \). Each eigenvector \( x \) satisfies the property \( x^* B x = 1 \). The eigenvectors are computed if options contains ComputeEigenvectors.

In addition, the two following variants can be solved via options:

• ABx_lx: \( ABx = \lambda x \)
• BAx_lx: \( BAx = \lambda x \)

Example:

MatrixXd X = MatrixXd::Random(5,5);
MatrixXd A = X + X.transpose();
cout << "Here is a random symmetric matrix, A:" << endl << A << endl;
X = MatrixXd::Random(5,5);
MatrixXd B = X * X.transpose();
cout << "and a random postive-definite matrix, B:" << endl << B << endl << endl;
 
GeneralizedSelfAdjointEigenSolver<MatrixXd> es(A,B);
cout << "The eigenvalues of the pencil (A,B) 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 A * v = " << endl << A * v << endl;
cout << "... and lambda * B * v = " << endl << lambda * B * v << endl << endl;

Output:

Here is a random symmetric 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
and a random postive-definite matrix, B:
  0.132  0.0109 -0.0512  0.0674  -0.143
 0.0109    1.68    1.13   -1.12   0.916
-0.0512    1.13     2.3   -2.14    1.86
 0.0674   -1.12   -2.14    2.69   -2.01
 -0.143   0.916    1.86   -2.01    1.68

The eigenvalues of the pencil (A,B) are:
  -227
  -3.9
-0.837
 0.101
  54.2
The matrix of eigenvectors, V, is:
   14.2   -1.03  0.0766 -0.0273   -8.36
 0.0546  -0.115   0.729   0.478   0.374
  -9.23   0.624 -0.0165   0.499    3.01
   7.88     1.3   0.225   0.109   -3.85
   20.8   0.805  -0.567 -0.0828   -8.73

Consider the first eigenvalue, lambda = -227
If v is the corresponding eigenvector, then A * v = 
 22.8
-28.8
 19.8
 21.9
-25.9
... and lambda * B * v = 
 22.8
-28.8
 19.8
 21.9
-25.9

See also

compute(const MatrixType&, const MatrixType&, int)

Computes generalized eigendecomposition of given matrix pencil.

Parameters

[in]	matA	Selfadjoint matrix in matrix pencil. Only the lower triangular part of the matrix is referenced.
[in]	matB	Positive-definite matrix in matrix pencil. Only the lower triangular part of the matrix is referenced.
[in]	options	A or-ed set of flags {ComputeEigenvectors,EigenvaluesOnly} | {Ax_lBx,ABx_lx,BAx_lx}. Default is ComputeEigenvectors|Ax_lBx.

Returns

Reference to *this

According to options, this function computes eigenvalues and (if requested) the eigenvectors of one of the following three generalized eigenproblems:

• Ax_lBx: \( Ax = \lambda B x \)
• ABx_lx: \( ABx = \lambda x \)
• BAx_lx: \( BAx = \lambda x \) with matA the selfadjoint matrix \( A \) and matB the positive definite matrix \( B \). In addition, each eigenvector \( x \) satisfies the property \( x^* B x = 1 \).

The eigenvalues() function can be used to retrieve the eigenvalues. If options contains ComputeEigenvectors, then the eigenvectors are also computed and can be retrieved by calling eigenvectors().

The implementation uses LLT to compute the Cholesky decomposition \( B = LL^* \) and computes the classical eigendecomposition of the selfadjoint matrix \( L^{-1} A (L^*)^{-1} \) if options contains Ax_lBx and of \( L^{*} A L \) otherwise. This solves the generalized eigenproblem, because any solution of the generalized eigenproblem \( Ax = \lambda B x \) corresponds to a solution \( L^{-1} A (L^*)^{-1} (L^* x) = \lambda (L^* x) \) of the eigenproblem for \( L^{-1} A (L^*)^{-1} \). Similar statements can be made for the two other variants.

Example:

MatrixXd X = MatrixXd::Random(5,5);
MatrixXd A = X * X.transpose();
X = MatrixXd::Random(5,5);
MatrixXd B = X * X.transpose();
 
GeneralizedSelfAdjointEigenSolver<MatrixXd> es(A,B,EigenvaluesOnly);
cout << "The eigenvalues of the pencil (A,B) are:" << endl << es.eigenvalues() << endl;
es.compute(B,A,false);
cout << "The eigenvalues of the pencil (B,A) are:" << endl << es.eigenvalues() << endl;

Output:

The eigenvalues of the pencil (A,B) are:
  0.0289
   0.299
    2.11
    8.64
2.08e+03
The eigenvalues of the pencil (B,A) are:
0.000481
   0.116
   0.473
    3.34
    34.6

See also

GeneralizedSelfAdjointEigenSolver(const MatrixType&, const MatrixType&, int)