However, once we consider the radius of the numerical range (the numerical range is a set of complex numbers associated with a matrix and the radius is the supremum in absolute value of this set) something unexpected (at least to me I suppose!) has happened. The first sections of the paper are largely algebraic and tend to revolve around the matrices involved, but with the introduction of supremum it makes sense that the arguments should tend towards analytical lines of thought. What is surprising is that the result comes about easily through a series of geometric observations on the set!
The graphic included is one I designed in LaTeX using the Tikz package, and it compactly encapsulates one of the proofs. The point of the result is that for any complex number not in the set we can find an appropriate shift such that the absolute value is strictly greater than the radius. While intuitive, the key here is finding the appropriate shift and our proof is surprisingly constructive.
At any rate, I really need to stop fooling around (read: looking for packages to print latex directly to images, giving up, and finally figuring out how to make photoshop do it without messing up the picture) and get the work I want to do done. Hopefully I can post a "primer" post soon and illuminate the subject of our research to the uninitiated!