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.
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