Statistical spectral analysis of random Gramian matrices

In this paper we perform the statistical analysis on the spectrum of random N × N Gramian matrices of the form G*_T Δ V^T · GT · V, where G and G_T are themselves Gramian matrices with subspace distance Δ(G,G_T) and V is the diagonalizer of G. In particular, we employ an extreme-value and asymptotic take on the theory of Gershgorin spectrum bounds to characterize the statistical structure of G*_T. The results reveal that even for relatively large Δ(G,G_T), the matrix G*_T is, with a high probability, brought to such a structure that can it be quickly diagonalized. This feature is exploited to design a statistically optimized and truncated variation of the Jacobi algorithm which is found to converge to the dominant eigenspace of G*_T as fast as the deterministic optimal sweeping strategy but without requiring its typical exhaustive search.