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 
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/
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Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

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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