On a conjecture in stability analysis of polytopic systems — Mimicking spectral radius properties from real to complex matrices

Our work discusses a conjecture formulated for supporting the stability analysis of polytopic systems, within the context of enlarging our explorations from real to complex statespace models. This conjecture refers to a property of the spectral radius that is true for real matrices and we assumed to be adaptable to complex matrices. Our presentation is organized as an extensive case study aiming to illustrate contradictory situations that may occur relative to a debatable mathematical statement, when neither a rigorous proof, nor counter-examples can be found. We show that the discussed conjecture was sustained by a systematic set of Monte Carlo tests, which (despite their generality) could not provide counter-examples and, subsequently, stimulated our assiduous search for a theoretical proof. Instead of the presumable proof, the theoretical investigations lead to some counter-examples, offered by a class of matrices with special structures. Eventually, we reevaluate the unreliable results obtained from the Monte Carlo tests and we explain the drawbacks of these tests yielding wrong conclusions.