Geodesic
On the relationship between multiplicities of the matrix spectrum and signs of components of its eigenvector in a~tree-like structure
Zapiski Nauchnykh Seminarov POMI, Combinatorics and graph theory. Part IX, Tome 464 (2017), pp. 5-25

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Tree-like structure parametric representation of an eigenspace corresponding to an eigenvalue $\lambda$ of a matrix $G$ is obtained in the case where a non-zero main basic minor of the matrix $G-\lambda E$ exists. If the algebraic and geometric multiplicities of $\lambda$ are equal, such a minor always exists. Coefficients at the degrees of spectral parameter are sums of summands having the same sign. If there is no non-zero main basic minor, the tree-like form does not allow to represent coefficients as sums with the same signs with the only exception – the case of eigenvalue of geometric multiplicity 1. 
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     author = {V. A. Buslov},
     title = {On the relationship between multiplicities of the matrix spectrum and signs of components of its eigenvector in a~tree-like structure},
     journal = {Zapiski Nauchnykh Seminarov POMI},
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     publisher = {mathdoc},
     volume = {464},
     year = {2017},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2017_464_a0/}
}
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V. A. Buslov. On the relationship between multiplicities of the matrix spectrum and signs of components of its eigenvector in a~tree-like structure. Zapiski Nauchnykh Seminarov POMI, Combinatorics and graph theory. Part IX, Tome 464 (2017), pp. 5-25. http://geodesic.mathdoc.fr/item/ZNSL_2017_464_a0/