An extension of the numerical range over finite fields

Abstract

Studying linear algebra over finite fields is fascinating since many properties of the real and complex numbers do not carry over directly to finite fields. In particular, the finite fields are not ordered, we do not have a notion of positivity, and we have a nonzero characteristic. Despite this, it is possible to develop interesting results related to an inner product like structure, norms, and numerical ranges. One of the challenges is that many nonzero vectors turn out to have a “norm” of zero. We give a formula for the number of vectors with norm zero, and analyze the effect that these zero norm vectors have in derailing some classical results over the real and complex numbers. In particular, we give examples of matrices for which the classical definition of a numerical range need not contain the eigenvalues of the matrix itself, and then extend the definition of numerical range and show that this new definition does indeed encompass all the eigenvalues.