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

By applying the bilinear transformation to an-variable polynomial, withn \geq 1, one arrives at a rational function, the zeros of which represent the "transformed polynomial." LetQbe the matrix relating the coefficients of the original polynomial to those of the transformed polynomial. It is shown that the matrixQhas a unique property, namely,Q^{2} = kl, wherekis a positive constant which is explicitly derived for the various cases discussed, andIis the unit matrix of the pertinent order.