## 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.

### MSC:

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

### References:

 [1] M. Avellaneda and J.-H. Lee. Statistical arbitrage in the US equities market. Quantitative Finance, 10:761-782, 2010. · Zbl 1194.91196 [2] A. Berman, M. Catral, L. De Alba, et al. Sign patterns that allow eventual positivity. Electronic Journal of Linear Algebra, 19:108-120, 2010. · Zbl 1190.15031 [3] P.P. Boyle. Positive weights on the efficient frontier. North American Actuarial Journal, 18:462-477, 2014. [4] P.P. Boyle, S. Feng, D. Melkuev, J. Zhang, and Y. Alex. Short Positions in the first principal component portfolio. North American Actuarial Journal, to appear, 2018. · Zbl 1393.91129 [5] A. Elhashash and D.B. Szyld. Two characterizations of matrices with the Perron-Frobenius property. Numerical Linear Algebra with Applications, 16:863-869, 2009. · Zbl 1224.15064 [6] D. Handelman. Positive matrices and dimension groups affiliated to c*-algebras and topological markov chains. Journal of Operator Theory, 6:55-74, 1981. · Zbl 0495.06011 [7] R.A. Horn and C.R. Johnson. Matrix Analysis, second edition. Cambridge University Press, Cambridge, 2013. [8] C.R. Johnson and P. Tarazaga. On matrices with Perron-Frobenius properties and some negative entries. Posi tivity, 8:327-338, 2004. · Zbl 1078.15018 [9] C.-K. Li and S. Pierce. Linear operators preserving correlation matrices. Proceedings of the American Mathe matical Society, 131:55-63, 2003. · Zbl 1006.15006 [10] A.M. Mercer and P.R. Mercer. Cauchy’s interlace theorem and lower bounds for the spectral radius. International Journal of Mathematics and Mathematical Sciences, 23:563-566, 2000. · Zbl 0955.15007 [11] D. Noutsos. On Perron-Frobenius property of matrices having some negative entries. Linear Algebra and its Applications, 412:132-153, 2006. · Zbl 1087.15024 [12] P. Tarazaga, M. Raydan, and A. Hurman. Perron-Frobenius theorem for matrices with some negative entries. Linear Algebra and its Applications, 328:57-68, 2001. · Zbl 0989.15008
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.