Non-Hermitian uncertainty principle for BGFT

Formulate an uncertainty principle for BGFT on directed graphs that (i) uses the Gram-metric G = V*V to define spectral-domain variance and (ii) captures dual localization properties of left and right eigenvectors, and relate the resulting trade-offs to conditioning measures such as the eigenbasis condition number kappa(V).

Background

Classical uncertainty principles rely on orthonormal bases and Euclidean metrics; these do not hold in non-Hermitian settings where biorthogonal bases and the Gram-induced metric govern energy and localization.

The paper proposes extending uncertainty concepts to the BGFT context by accounting for asymmetric geometry and conditioning, thereby providing foundational limits for simultaneous localization in node and spectral domains of directed, non-normal graphs.