MatrixXd A = MatrixXd::Random(6,6);
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cout << "Here is a random 6x6 matrix, A:" << endl << A << endl << endl;
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EigenSolver<MatrixXd> es(A);
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cout << "The eigenvalues of A are:" << endl << es.eigenvalues() << endl;
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cout << "The matrix of eigenvectors, V, is:" << endl << es.eigenvectors() << endl << endl;
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complex<double> lambda = es.eigenvalues()[0];
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cout << "Consider the first eigenvalue, lambda = " << lambda << endl;
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VectorXcd v = es.eigenvectors().col(0);
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cout << "If v is the corresponding eigenvector, then lambda * v = " << endl << lambda * v << endl;
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cout << "... and A * v = " << endl << A.cast<complex<double> >() * v << endl << endl;
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MatrixXcd D = es.eigenvalues().asDiagonal();
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MatrixXcd V = es.eigenvectors();
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cout << "Finally, V * D * V^(-1) = " << endl << V * D * V.inverse() << endl;
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