Exact solution for the rank-one structured singular value with repeated complex full-block uncertainty

In this note, we present an exact solution for the structured singular value (SSV) of rank-one complex matrices with repeated complex full-block uncertainty. A key step in the proof is the use of Von Neumann's trace inequality. Previous works provided exact solutions for rank-one SSV when the uncertainty contains repeated (real or complex) scalars and/or non-repeated complex full-block uncertainties. Our result with repeated complex full-blocks contains, as special cases, the previous results for repeated complex scalars and/or non-repeated complex full-block uncertainties. The repeated complex full-block uncertainty has recently gained attention in the context of incompressible fluid flows. Specifically, it has been used to analyze the effect of the convective nonlinearity in the incompressible Navier–Stokes equation (NSE). SSV analysis with repeated full-block uncertainty has led to an improved understanding of the underlying flow physics. We demonstrate our method on a turbulent channel flow model as an example.