The nice thing about circulant matrices is that their spectrum and eigenvectors can be computed explicitly. This can be used in combinatorics, for instance, to compute the spectrum of Cayley graphs of cyclic groups.

Thank you very much for your interesting answer :-)

 as patrick sais, you can compute all eigenvalues and eigenvectors, regardless  of

size from matrices in this category (one of the few examples) also called cyclic matrices.

If one takes a linear chain of atoms in the so called tight binding theory, with no disorder,

you get the spectrum exactly.

any two matrices of this type, for different parameters, commute. among themselvs, for same size matrices. therefore you get  commutative ring, with equivalent algebra as the eigenvalues. the eigenvalues could be considered commutative  ring numbers.

the 2 by 2 case

x  y

y  x

give x+ty, tt=1, the hyperbolic or perplex system, whole system in itself similar to complex.

Functions over these numbers then retain the same form, the two components obey

the 1D wave equation, in the same sense that components of a complex function

obey the laplace equation.

You get three systems simultaneously by taking

x  y

ty  x

or x+ty for tt=-1,0,1

ring of dual nos. for tt=0