Fail to compute eigenvalues for a simple 3x3 companion matrix for root finding

Submitted by Gary Tan

Assigned to Nobody

Link to original bugzilla bug (#1557)
Version: 3.3 (current stable)

Description

The following

#include <Eigen/Eigenvalues>

#include <iostream>

using namespace Eigen;

using namespace std;

int main() {

Matrix3d A;

A << 0, 0, 0, 1, 0, 0.5887907064808635127, 0, 1, 0;

cout << "Here is a random 3x3 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&lt;double&gt; >() * v << endl  

   << endl;

MatrixXcd D = es.eigenvalues().asDiagonal();

MatrixXcd V = es.eigenvectors();

cout << "Finally, V * D * V^(-1) = " << endl << V * D * V.inverse() << endl;

return 0;

}

produces

Here is a random 3x3 matrix, A:

   0        0        0  

   1        0 0.588791  

   0        1        0

The eigenvalues of A are:

(0,0)

The matrix of eigenvectors, V, is:

(1,0) (0,0) (0,0)

(0,0) (0,0) (0,0)

Consider the first eigenvalue, lambda = (0,0)

If v is the corresponding eigenvector, then lambda * v =

(0,0)

... and A * v =

(0,0)

(1,0)

(0,0)

Finally, V * D * V^(-1) =

(-nan,-nan) (-nan,-nan) (-nan,-nan)

However, it's expected to obtain

sqrt(0.5887907064808635127) = 0.76733

and its opposite.

Thanks,

Gary

Blocking

#814