Motivated by decentralized-control and numerical-methods applications, we explore the scaling problem. That is, we consider the design of a diagonal matrix K so as to (i) place the eigenvalues of the product KG in the OLHP and (ii) optimize a scalar performance measure c(KG) of interest, for a square topology matrix G. We first illustrate our methodology for scaling design, which combines optimization machinery with graph-algebra notions, for a class of topology matrices and a specific performance measure. In particular, for symmetric positive definite topologies and for a dominant-eigenvalue-ratio performance measure, we find that the optimally-scaled system KG has a special structure: the right eigenvectors associated with the largest and smallest eigenvalues have corresponding entries of identical magnitude and hence differ only in sign pattern. Using this structure, we develop both explicit and algorithmic methods for computing the optimal scaling and cost. After illustrating our methodology with this example, we characterize the optimally-scaled system for several relevant cost measures and classes of topology matrices. Finally, we give some first explorations on how these methods can be extended to address the more general eigenvalue-placement problem.
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