D.4 Why are complex numbers important?

At first glance, complex numbers might not seem so important in machine learning. Let me assure you that this is not the case: they are absolutely essential. In this section, we will take a quick look forward and see what complex numbers make possible.

For instance, one of the most important applications of complex numbers is the eigenvalue-eigenvector pairs of matrices. For a given matrix A, there are special numbers λ called eigenvalues, and corresponding vectors v called eigenvectors, such that Av = λv holds. In the language of linear transformations, this means that in the linear subspace spanned by the eigenvectors of λ, the transformation A is just a stretching.

These are very powerful, as under certain conditions, eigenvalues allow us to simplify the matrix.

Certain square real matrices can be written in the form

is a diagonal matrix composed of the eigenvalues of A. Guess what guarantees the existence of eigenvalues...