Elementary symmetric functions of two solvents of a quadratic matrix equation

Quadratic matrix equations occur in a variety of applications. In this paper we introduce new permutationally invariant functions of two solvents of the n × n quadratic matrix equation X2 -ℒ1X -ℒ0 = 0, playing the role of the two elementary symmetric functions of the two roots of a quadratic scalar equation. Our results rely on the connection existing between the QME and the theory of linear second-order difference equations with noncommutative coefficients. An application of our results to a simple physical problem is briefly discussed.