On the Relationship Between the Multiplicities of the Matrix Spectrum and the Signs of the Components of its Eigenvectors in a Tree-Like Structure
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We obtain a tree-like parametric representation of the eigenspace corresponding to an eigenvalue ⋋ of a matrix G in the case where the matrix G − ⋋E has a nonzero principal basic minor. If the algebraic and geometric multiplicities of ⋋ coincide, then such a minor always exists. The coefficients of powers of the spectral parameter are sums of terms of the same sign. If there is no nonzero principal basic minor, then the tree-like form does not allow one to represent the coefficients as sums of terms of the same sign, the only exception being the case of an eigenvalue of geometric multiplicity 1.
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