Elementary symmetric functions of two solvents of a quadratic matrix equations

Quadratic matrix equations occur in a variety of applications. Inthis paper we introduce new permutationally invariant functions oftwo solvents of the n £ n quadratic matrix equation X2 ¡ L1X ¡L0 = 0, playing the role of the two elementary symmetric functionsof the two roots of a quadratic scalar equation. Our results rely onthe connection existing between the QME and the theory of linearsecond order di®erence equations with noncommutative coe±cients.An application of our results to a simple physical problem is brie°ydiscussed.