Involutional Matrices Based on the Representation Theory of GL(2)

Involutional matrices M(a, b, c) with three arbitrary parameters are introduced, based on a matrix representation M(R) of the linear homogeneous transformation R ∈ GL(2). Symmetry properties, eigenvalues, and recursion formulas for the representation M(R) are obtained and specialized to the involutional matrices M(a, b, c). A set of special involutional matrices A(ξ), B(ξ), C(ξ), and E(ξ) with one arbitrary parameter ξ are introduced as special cases of M(a, b, c). Their relations are discussed.