Although the state space or the internal description of the system still involves a relationship between the input and output signals, it also involves an additional set of variables, called State variables.

We define the state of a system at time n0 as the amount of information that must be provided at time n0, which, together with the input signal x(n) for n≥n0 determines output signal for n≥n0.

If h(n) is the impulse response of the single input-single output system, and h1(n) is the impulse response of the transposed system, then we know that h(n)=h1>(n). Thus, a single input-single output system and its transpose have identical impulse responses and hence the same input-output relationship.

The parallel form realization is also known as normal form representation, because the matrix F is diagonal, and hence the state variables are uncoupled.

According to the definition of state of a system, the system consists of two components called memory component and memory less component.

A closed form solution of the state space equations is easily obtained when the system matrix F is diagonal. Hence, by finding a matrix P so that F1=PFP-1 is diagonal, the solution of the state equations is simplified considerably.

The transpose of the matrix F is obtained by interchanging its rows and columns, and it is denoted by FT. The system thus obtained is known as Transposed system.

The state variables provide information about all the internal signals in the system. As a result, the state-space description provides a more detailed description of the system than the input-output description.

A number λ is an Eigen value of F and a nonzero vector U is the associated Eigen vector if

We know that (F-λI)U=0