  In engineering applications, one of the major challenges today is to develop
reliable and robust control algorithms for complex networked systems.
Controllability and observability of such systems play a crucial role in the
design process. The underlying network structure may contain symmetries --
caused for example by the coupling of identical building blocks -- and these
symmetries lead to repeated eigenvalues in a generic way. This complicates the
design of controllers since repeated eigenvalues might decrease the
controllability of the system. In this paper, we will analyze the relationship
between the controllability and observability of complex networked systems and
graph symmetries using results from representation theory. Furthermore, we will
propose an algorithm to compute sparse input and output matrices based on
projections onto the isotypic components. We will illustrate our results with
the aid of two guiding examples, a network with $ D_4 $ symmetry and the
Petersen graph.
