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A linear time-invariant discrete-time system is described by the vector matrix difference equation $$ x(k+1)=F \underline{X}(k)+G \underline{u}(k) $$

Where $\underline{X}(k)$ is the state vector, $F$ is an $n \times n$ constant matrix, $G$ is a $(n \times r)$ constant matrix and $\underline{u}(k)$ is the control vector. The state transition matrix of the system is given by inverse $Z$-transform of

- $ZI - F$
- $(Z I-F) Z$
- $(Z I-F)^{-1} G$
- $(Z I-F)^{-1} Z$