ELA, Volume 16, pp. 125-134, May 2007, abstract.

A Generalization of Rotations and Hyperbolic Matrices 
and Its Applications

M. Bayat, H. Teimoori, and B. Mehri

In this paper, A-factor circulant matrices with the 
structure of a circulant, but with the entries below the 
diagonal multiplied by the same factor A are introduced. 
Then the generalized rotation and hyperbolic matrices are
defined, using an idea due to Ungar. Considering the 
exponential property of the generalized rotation and 
hyperbolic matrices, additive formulae for corresponding
matrices are also obtained. Also introduced is the block 
Fourier matrix as a basis for generalizing the Euler formula.
The special functions associated with the corresponding Lie 
group are the functions F^A_{n,k}(x) (k=0,1,...,n-1). As an 
application, the fundamental solutions of the second order 
matrix differential equation y''(x)=Pi_A y(x) with initial 
conditions y(0)=I and y'(0)=0 are obtained using the 
generalized trigonometric functions cos_A(x) and sin_A(x).