Pseudospectral Bounds and Transient Amplification in Low-Rank Economic Systems

Pseudospectral analysis provides a natural framework for assessing robustness, systemic vulnerability, and transient shock amplification in non-normal dynamical systems, but its direct application becomes computationally demanding in large-scale problems. Recently, a scalable framework for matrices of the form A=UV*, with U,V∈Cd×r and r≪d, was developed, and the computation of the smallest singular value of zI−A, together with several related pseudospectral problems, was reduced to structured eigenvalue problems whose dimensions depend only on the low rank r. Building on this framework, the present paper develops further theoretical and computational results motivated by systemic-risk analysis in large-scale economic and financial interaction models. We extend the analysis to diagonally shifted low-rank systems and derive a characterization of pseudospectral boundary intersections with circular trajectories centered at the diagonal shift. For approximately low-rank matrices, known localization results are used to derive computable lower and upper bounds for the distance to instability. As a new theoretical contribution, we establish an analytical lower bound for the smallest eigenvalue of the reduced matrix MU,V(z), together with computable refinements involving only reduced-order quantities; the bound is shown to be sharp in a representative example. The reduced framework is further employed in the computation of the Kreiss constant and the analysis of transient amplification. Numerical experiments on large-scale economic interaction models illustrate how these results enable computationally tractable assessment of robustness and transient risk through reduced problems whose dimensions are independent of the full system size.