A general property of the transformation matrices associated with the n-variable bilinear transformation

By applying the bilinear transformation to a n -variable polynomial, with n \geq 1 , one arrives at a rational function, the zeros of which represent the "transformed polynomial." Let Q be the matrix relating the coefficients of the original polynomial to those of the transformed polynomial. It is shown that the matrix Q has a unique property, namely, Q^{2} = kl , where k is a positive constant which is explicitly derived for the various cases discussed, and I is the unit matrix of the pertinent order.