ON THE VARIANCE OF EIGENVALUES OF THE COMMUNITY MATRIX: DERIVATION AND APPRAISAL

Abstract

Eigenvalues, the solutions to the characteristic polynomial, are important measures of community behavior. Their range and practical measurement present difficult challenges in ecology. We therefore present the derivation of variance of eigenvalues of the community matrix, var(λ) = var (aii) + (n − 1)aijaji, as well as a novel related formula, namely, the expectancy of pairwise eigenvalues (EPV), var(λpairwise) = var(aii−pairwise) + aijaji. We propose that the two formulae may be useful in evaluating the relative contributions of inter- and intraspecific effects on the behavior of large systems. EPV allows estimating eigenvalue distribution of systems of unknown size.