Correlation matrices with the Perron Frobenius property. (English) Zbl 1390.15026

Summary: 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.


15A18 Eigenvalues, singular values, and eigenvectors
15B48 Positive matrices and their generalizations; cones of matrices
91G10 Portfolio theory


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