Self-adjoint matrix equations on time scales

In this study, linear second-order delta-nabla matrix equations on time scales are shown to be formally self-adjoint equations with respect to a certain inner product and the associated self-adjoint boundary conditions. After a connection is made with symplectic dynamic systems on time scales, we introduce a generalized Wronskian and establish a Lagrange identity and Abel's formula. Two reduction-of-order theorems are given. Solutions of the second-order self-adjoint equation are then shown to be related to corresponding solutions of a first-order Riccati equation. Then a comprehensive roundabout theorem relating key equivalences is stated. Finally several oscillation theorems are proven about the self-adjoint equation. We then go on to state similar results for the nabla-delta matrix equation.