Perron Frobenius theorem for matrices with entries from $\{-1,0,1\}$ 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.
 A: Garrett Birkhoff showed that if an operator is a strict contraction of a proper convex real cone $C$ then you obtain the same conclusions as in the Perron-Frobenius Theorem. In the finite dimensional case of a matrix, you get that $\lambda=\rho_{\rm sp}(M)>0$ is a simple eigenvalue associated with an eigenvector $h\in C^*=C\setminus\{0\}$ and all other eigen-values are strictly smaller in modulus. The standard P-F theorem is for $C={\Bbb R}^n_+$ for which a matrix $M=(m_{ij})$ with all $m_{ij}>0$ maps $C^*$ into the interior of $C$, which suffices to conclude. [Unfortunately, I don't have any wiki page or easy reference on all this].
This may not give you much help, however, as it only works when you really can find such a cone. I had to go to 5 by 5 matrices to find an example with $\pm 1$ entries:
$$M=\pmatrix{-1 & 1 & 1 & 1 & 1 \\ 1 & -1 & 1 & 1 & 1\\ 1 & 1 & -1 &1 & 1
\\1 & 1 & 1 & -1 & 1\\ 1 & 1 & 1 & 1 & -1}$$
The important part here is that in all rows (or columns) you have 4 ones and only one -1.
Let $e$ be the vector of all ones. You then have $Me=3e$ while if $(e,z)=\sum_{i=1}^5 x_i =0$ then $|Mz|_1 \leq 2 |z|_1$. Because 2 is smaller than 3, $M$ is a strict contraction of the cone (leaving details to the reader):
$$C = \left\{ x\in {\Bbb R}^5: \left|x-\frac{1}{5}e (e,x) \right|_1 \leq (e,x) \right\}.$$
Thus by Birkhoff, this matrix  has a simple largest eigenvalue (here 3) and all other eigenvalues are strictly smaller in modulus (here not greater than 2). For the similar 4 by 4 example with the last row and column removed, the conclusion is false: The leading eigenvalue is double.
A: 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:

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*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:

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*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:

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*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.)
A: It is known that the Perron-Frobenius theorem no longer holds when negative entries are allowed.
See Belardo, Francesco, et al. "Open problems in the spectral theory of signed graphs." arXiv preprint arXiv:1907.04349 (2019).
https://arxiv.org/abs/1907.04349
In particular, the absolute value of the smallest eigenvalue may be larger than the largest eigenvalue. This leads to some interesting open problems.
The fact that the eigenvector corresponding to the largest eigenvalue may have negative entries is easily seen by similarity transformations.
