NON-NORMALITY SHAPES AAA RATIONAL APPROXIMATION OF OPERATORS: A DE-NONNORMALIZATION PROBE

This work introduces a diagnostic probe for AAA rational approximation by comparing the AAA fit of an operator with that of its symmetrized surrogate. The probe reveals that non-normality—particularly one-sided triangular coupling—induces a phase transition in the single-moment residue utilization across a sharp threshold, and the symmetrized control reverses this effect. De-nonnormalization removes spurious poles and excess residue mass while preserving the genuine spectral ridge geometry. Furthermore, we reveal that the underlying mechanism is geometric: non-normality anisotropically expands the pseudospectral boundary, forcing AAA to spend extra poles to cover the expanded region. Results hold on synthetic matrices and generalize to brain connectomes and ecological networks. Finally, building on these findings, we propose a possible geometric approach to proving AAA effectiveness, reducing the problem to pole coverage of the pseudospectrum.