Transposition of a State Space Model

Above, we found the transfer function of the general state-space model to be

$$H(z) = D + C \left(zI - A\right)^{-1}B.$$

By the rules for transposing a matrix, the transpose of this equation gives

$$H^T(z) = D^T + B^T \left(zI - A^T\right)^{-1}C^T.$$

The system $(A^T,C^T,B^T,D^T)$ may be called the transpose of the system $(A,B,C,D)$. The transpose is obtained by interchanging $B$ and $C$ in addition to transposing all matrices.

When there is only one input and output signal (the SISO case), $H(z)$ is a scalar, as is $D$. In this case we have

$$H(z) = D + B^T \left(zI - A^T\right)^{-1}C^T.$$

That is, the transfer function of the transposed system is the same as the untransposed system in the scalar case. It can be shown that transposing the state-space representation is equivalent to transposing the signal flow graph of the filter [72]. The equivalence of a flow graph to its transpose is established by Mason's gain theorem [49,50]. See §9.1.3 for more on this topic.