The Eigenvalues of Mega-dimensional Matrices

Abstract

Often, we need to know some integral property of the eigenvalues {x} of a large N × N symmetric matrix A. For example, determinants det (A) = exp(∑ log (x)) play a role in the classic maximum entropy algorithm [Gull, 1988] . Likewise in physics, the specific heat of a system is a temperature- -dependent sum over the eigenvalues of the Hamiltonian matrix. However, the matrix may be so large that direct O (N ³ calculation of all N eigenvalues is prohibited. Indeed, if A is coded as a “fast” procedure, then O (N ² operations may also be prohibited.

Then the only permitted use of A is to apply it to one or a few vectors v₀, v₁, v₂, .... We use the resulting vectors in an entropic Bayesian algorithm to estimate the eigenvalue spectrum of A, and thence its integral properties. A million-by-million matrix is used as an example.