Rayleigh Quotient as a Measure of Informational Energy: Analytical and Geometric Characterization of Equilibrium States

The Rayleigh Quotient (RQ), classically defined as $R_Q = \frac{\vb{A}^T \vb{G} \vb{A}}{\vb{A}^T \vb{A}}$, plays a fundamental role in the theory of vibrations, eigenvalue problems, and stability analysis. In this work, we reinterpret $R_Q$ as a measure of \emph{informational energy} --- a scalar quantifying the degree of geometric coherence between informational flow (represented by a vector $\vb{A}$) and a constraining geometry $\vb{C}$ defining the manifold $\vb{G} \vb{C} \vb{G}^T = \vb{I}$. We show analytically and numerically that the equilibrium value of the quotient, $R_Q^* = \vb{A}^T \vb{C}^{-1} \vb{A}$, corresponds to the informational steady state, representing maximal geometric harmony between the system's internal curvature and its external informational drive. The presented framework connects spectral geometry, constraint optimization, and informational physics, offering a unified language for interpreting stability and coherence in complex systems.