 Three-coordinate state realization of an ARMA(1,1) process

For an autoregressive moving-average model of order $(1,1)$, this state has observation row $F=(\varphi ,1,\theta )$ and transition matrixThe state noise is $(0,{\epsilon }_t,0{)}^{\top }$. Its first transition row reconstructs $X_{t-1}$ and its third row shifts the previous white noise value.