On the relationship between multiplicities of the matrix spectrum and signs of components of its eigenvector in a tree-like structure

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.