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).