Perron-Frobenius theory, Correlation matrix, Positive eigenvector
This paper investigates conditions under which correlation matrices have a strictly positive dominant eigenvector. The sufficient conditions, from the Perron-Frobenius theorem, are that all the matrix entries are positive. The conditions for a correlation matrix with some negative entries to have a strictly positive dominant eigenvector are examined. The special structure of correlation matrices permits obtaining of detailed analytical results for low dimensional matrices. Some specific results for the $n$-by-$n$ case are also derived. This problem was motivated by an application in portfolio theory.
Boyle, Phelim P. and N'Diaye, Thierno B..
"Correlation Matrices with the Perron Frobenius Property",
Electronic Journal of Linear Algebra,
Volume 34, pp. 240-268.