Some mathematical topics, circulant matrices, in particular, are pure gems that cry out to be admired and studied with different techniques or perspectives in mind. Circulant matrices are prevalent in many parts of mathematics (see, for example, [8]). We point the reader to the elegant treatment given in [4, §5.2], and to the monograph [1] devoted to the subject. These matrices appear naturally in areas of mathematics where the roots of unity play a role, and some of the reasons for this to be so will unfurl in our presentation. However ubiquitous they are, many facts about these matrices can be proven using only basic linear algebra. This makes the area quite accessible to undergraduates looking for research problems, or mathematics teachers searching for topics of unique interest to present to their students.
We concentrate on the discussion of necessary and sufficient conditions for circu- lant matrices to be non-singular, and on various distinct representations they have, goals that allow us to lay out the rich mathematical structure that surrounds them. Our treatement though is by no means exhaustive. We expand on their connection to the algebraic geometry over a field with one element, to normal curves, and to Toeplitz’s operators. The latter material illustrates the strong presence these matrices have in various parts of modern and classical mathematics. Additional connections to other mathematics may be found in [11].
Professor Kra's paper can be found here.