A generalization of rotation and hyperbolic matrices and its applications

Abstract

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
correspondingmatrices are also obtained. Also introduced is the block Fourier matrix as a basis for
generalizingthe Euler formula. The special functions associated with the corresponding Lie group
are the functions FA_n,k(x) (k = 0, 1, ··· , n − 1). As an application, the fundamental solutions of
the second order matrix differential equation y"(x)=Π_Ay(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)