# Finding Complex Eigenvalues of Hessenberg Matrix

Given a Hessenberg matrix, i wish to compute its eigenvalues using QR Algorithm.

The problem is that the matrix has complex eigenvalues and my implementation of the QR Algorithm can't find them.

It sucessfully finds the eigenvalues that have only real part, but the matrix never finds complex eigenvalues.

How can I proceed? Is there a version of QR Algorithm that does it? Is there a paper or book that may help me?

Example of matrix:

array([[-4.64628998e+03+0.j, -5.68688457e+02+0.j,  2.71866272e+02+0.j,
-7.00797089e+01+0.j, -4.64131909e+02+0.j],
[ 3.75304668e+04+0.j,  4.64667849e+03+0.j, -2.45500711e+03+0.j,
3.80887022e+02+0.j,  4.00162200e+03+0.j],
[ 0.00000000e+00+0.j,  5.26447278e+02+0.j, -5.54565702e+01+0.j,
-8.12474922e+01+0.j,  6.47352607e+01+0.j],
[ 0.00000000e+00+0.j,  0.00000000e+00+0.j,  2.77255195e+03+0.j,
6.41617510e+01+0.j, -2.95459594e+03+0.j],
[ 0.00000000e+00+0.j,  0.00000000e+00+0.j,  0.00000000e+00+0.j,
5.14603305e+02+0.j, -1.65955603e+01+0.j]])

• I would prefer having the code, and implementing it. – arthurp36 Jan 9 at 18:22
• i am using python – arthurp36 Jan 9 at 18:36