When finding eigenpairs of some matrices one may encounter an imaginary numbers. For example, given the matrix
One attempts to first find the eigenvalues of the matrix:
As shown above the square of any number cannot be negative, thus the above equality is impossible.
So in order to find the eigenpairs of the above matrix one must use complex numbers. First we build our toolkit. We observe Euler's Formula, the complex conjugate rule, and the quadratic formula:
Using this our tool kit we can now find the eigenvalues:
Now with the eigenvalues we are free to find the eigenvectors of the matrix as we would with a normal matrix:
For
See 1- Introduction Sect 4.6 Complex Eigenvalues Part 1 for more info.