  In the present paper we consider controllability and observability of second
order linear time invariant systems in matrix form. Without reducing into first
order systems we show how the classical conditions for first order linear
systems can be generalized to this case. In term of Kalman type criterions
these concepts are investigated for second order discrete and continuous time
linear systems. It should be pointed out that by repeated differentiation of
state and output vector-functions we derive two different systems of linear
algebraic equations. Then the initial values x_0, x_1 and input functions can
be determined uniquely from these systems if and only if the observability and
controllability matrices have full rank, respectively. Also the transfer
function of the second order continuous-time linear state-space system is
constructed. A numerical example is given to illustrate the feasibility and
effectiveness of the theoretic results obtained.
