This paper consists of two parts. First, the rational iterations obtained from certain Padé approximations associated with computing the matrix sign function are shown to be equivalent to iterations involving the hyperbolic tangent and its inverse. Using this equivalent formulation many results about these Padé iterations, such as global convergence, semigroup property, and partial fraction decompositions can be obtained easily. In the second part of the paper it is shown that the behaviour of points under the Padé iterations can be expressed, via Cayley transform, as the combined result of a completely regular iteration and a chaotic iteration. Thereby, the chaotic iteration takes the form of a multiplicative linear congruential random number generator.