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ON THE VARIANCE OF EIGENVALUES OF THE COMMUNITY MATRIX: DERIVATION AND APPRAISAL
Author(s) -
Jorgensen Jane,
Rossignol Annette MacKay,
Puccia Charles J.,
Levins Richard,
Rossignol Philippe A.
Publication year - 2000
Publication title -
ecology
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.144
H-Index - 294
eISSN - 1939-9170
pISSN - 0012-9658
DOI - 10.1890/0012-9658(2000)081[2928:otvoeo]2.0.co;2
Subject(s) - pairwise comparison , eigenvalues and eigenvectors , variance (accounting) , mathematics , intraspecific competition , range (aeronautics) , statistics , community , polynomial , matrix (chemical analysis) , ecology , econometrics , mathematical analysis , biology , economics , ecosystem , physics , materials science , accounting , quantum mechanics , composite material
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 ( a ii ) + ( n − 1) a ij a ji , as well as a novel related formula, namely, the expectancy of pairwise eigenvalues (EPV), var(λ pairwise ) = var( a ii −pairwise ) + a ij a ji . 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.