Above, we found the transfer function of the general state-space model to be
By the rules for transposing a matrix, the transpose of this equation gives
The system may be called the transpose of the system . The transpose is obtained by interchanging and in addition to transposing all matrices.
When there is only one input and output signal (the SISO case), is a scalar, as is . In this case we have
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 [75]. 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.