Source-Resolved Collective Cognition in Robotics: Mathematical Foundations for Multimodal and Collective Estimation, and Their Reach across the Cognition Loop

Abstract

The Generalized Singular Value Decomposition (GSVD) is a central tool for comparing two linear operators that act on the same unknown. Once the output metrics are fixed, it answers the basis-independent question "along which state directions does the first operator dominate the second, and by how much?". This foundational volume builds the GSVD and its surrounding estimation theory from the ground up. Part I re-derives every piece of linear algebra and statistical estimation machinery needed later -- inner products, the four subspaces, orthogonal matrices, the spectral theorem, quadratic forms, the ordinary SVD, Lie-group tangent perturbations, Jacobians, covariance, whitening, Fisher information, CCA, mutual information, discrimination criteria, and covariance intersection -- with self-contained derivations and worked numerical examples. Part II states the GSVD in both a transparent and a dimensionally general form, then proves it by two complementary routes. Part III works one small matrix pair through five different lenses, each computed by hand and verified numerically. Part IV separates information attribution, subspace geometry, statistical dependence, and decision-theoretic discrimination for collective systems. Part V evaluates the role and limits of the GSVD across the cognition loop. Part VI covers numerical practice, including a tested reference implementation. Every number printed in this document was verified in double precision. Where a statement is an assumption rather than a theorem, it is labelled as such.