Symmetric square roots of the infinite identity matrix

Abstract

Some non-trivial real, symmetric square roots of the infinite identity matrix are exhibited. These may be found either from the use of involutory integral transforms and a set of real orthonormal functions or by an algebraic factorisation procedure. The two approaches are shown to be equivalent. Guinand (1956) has shown that square roots of the infinite identity matrix I can be generated from a Fourier kernel using biorthogonal functions. In particular, he gives an explicit expression for a family of asymmetric square roots of I, an example which generalises an earlier result of Barrucand (1950).In this paper we use Guinand's analytical method to develop families of real, symmetric (and hence orthogonal) square roots of I, in the sequel referred to simply as "symmetric roots". We show that all such roots can be generated by this method, and that the method is equivalent .to an algebraic factorisation procedure. Some examples and applications are also presented.A number of texts on matrices give results on square roots of finite matrices (see, for example, Perlis (1952) Theorem 9-15), and indeed there is a literature on unilateral (polynomial) equations in a finite square matrix (MacDuffee (1946) Chapter 8 gives a bibliography). Nevertheless, apart from the work of Guinand noted above, scant attention seems to have been paid to square roots (or other fractional powers) of the infinite identity matrix. In his book on infinite matrices, Cooke (1950) mentions only some trivial examples as an exercise.Square roots of I and orthogonal matrices, or rather the associated transformations, have found application in connection with summation formulae and the corresponding inversion formulae (see Guinand (1939) and Smith (1972)).