14.1.3 Complex Eigenvalues
What about complex eigenvalues? If your matrix is real, you won’t see these by graphing the
characteristic equation on your calculator. Will the shifted inverse power method find these
eigenvalues and their associated eigenvectors? The answer is yes. However, for a real matrix, you
must pick α to be complex. This is because the eigenvalues occur in conjugate pairs so if you don’t
pick it complex, it will be the same distance between any conjugate pair of complex numbers and
so nothing in the above argument for convergence implies you will get convergence
to a complex number. Also, the process of iteration will yield only real vectors and
Example 14.1.7 Find the complex eigenvalues and corresponding eigenvectors for the
Here the characteristic equation is λ3 − 5λ2 + 8λ− 6 = 0. One solution is λ = 3. The other two
are 1 + i and 1 − i. I will apply the process to α = i to find the eigenvalue closest to
Then let u1 =
for lack of any insight into anything better.
Now divide by the largest entry to get the next iterate. This yields for an approximate eigenvector
Now leaving off extremely small terms,
so it appears that an eigenvector is the above and an eigenvalue can be obtained by
The method has successfully found the complex eigenvalue closest to i as well as the eigenvector.
Note that I used essentially 20 iterations of the method.
This illustrates an interesting topic which leads to many related topics. If you have a
polynomial, x4 + ax3 + bx2 + cx + d, you can consider it as the characteristic polynomial of a
certain matrix, called a companion matrix. In this case,
The above example was just a companion matrix for λ3 − 5λ2 + 8λ − 6. You can see the pattern
which will enable you to obtain a companion matrix for any polynomial of the form
λn + a1λn−1 +
. This illustrates that one way to find the complex zeros of a
polynomial is to use the shifted inverse power method on a companion matrix for the polynomial.
Doubtless there are better ways but this does illustrate how impressive this procedure is. Do you
have a better way?
Note that the shifted inverse power method is a way you can begin with something close but
not equal to an eigenvalue and end up with something close to an eigenvector.