Perron Frobenius theorem for matrices with entries from $\{-1,0,1\}$

eigenvalues-eigenvectorslinear algebramatricesreference-request

The Perron-Frobenius theorem is a well known theorem for positive symmetric matrices and irreducible non-negative matrices (it gives information about the largest eigenvalue and the existence of a positive/non-negative eigenvector corresponding to it).

I am looking for some reading material on the Perron-Frobenius theory for matrices with negative entries also. Particularly, is there anything known for symmetric matrices having entries $\{-1,0,1\}$?

EDIT: specific results on the existence of Perron like eigenvector, or if there is any nice characterization of the eigenvector(s) corresponding to the eigen value of maximum modulus, would be useful.

Best Answer

There is actually a whole theory about Perron-Frobenius theory for matrices whith some negative entries, which is closely related to so-called eventually non-negative and eventually positive matrices. This theory has been continuously growing for the last 20 years or so. Here are just a few references:

  • Charles R. Johnson, Pablo Tarazaga: On matrices with Perron–Frobenius properties and some negative entries, Positivity 8, No. 4, 327-338, 2004 (link zo zbMATH).

  • Dimitrios Noutsos: On Perron-Frobenius property of matrices having some negative entries, Linear Algebra Appl. 412, No. 2-3, 132-153, 2006 (link to zbMATH).

  • Dimitrios Noutsos, Michael J. Tsatsomeros: Reachability and holdability of nonnegative states, SIAM J. Matrix Anal. Appl. 30, No. 2, 700-712, 2008 (link to zbMATH).

On a related note, there is also a lot of literature on eventual positivity of sign patterns. Here's just one example:

  • Ber-Lin Yu, Ting-Zhu Huang, Cui Jie, Chunhua Deng: Potentially eventually positive star sign patterns, Electron. J. Linear Algebra 31, 541-548, 2016 (link to zbMATH).

Following the references and citations of some of the aforementioned papers, or having a look at the publication lists of their authors, will certainly give you many more papers on this topic.

A quite extensive (though certainly not comprehensive) list of references is also discussed in the paragraph On the history of eventual positivity in the introduction to the following paper by myself:

  • Towards a Perron-Frobenius theory for eventually positive operators, J. Math. Anal. Appl. 453, No. 1, 317-337, 2017 (link to zbMATH).

(The paper itself is about the infinite dimensional case, but many of the references discussed in said paragraph deal with the finite-dimensional case.)

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