On the eigenvectors belonging to the minimum eigenvalue of an essentially nonnegative symmetric matrix with bipartite graph
Signs of coordinates of eigenvectors belonging to the minimum eigenvalue m(B) of an essentially nonnegative symmetric matrix B whose graph G(B) is connected and bipartite are studied. As the main result the following theorem is proved: If G(B) is connected and if \(V_ 1{\dot \cup}V_ 2\) is a proper bipartition of the vertex set of G(B), then there exists an eigenvector x belonging to m(B) such that \(x_ i>0\) for \(i\in V_ 1\) and \(x_ i<0\) for \(i\in V_ 2\). The main technique is the use of the Rayleigh-quotient characterization of m(B). Using this theorem (and 2 other propositions on essentially nonnegative symmetric n-by-n matrices) the theorem by \textit{G. Constantine} [ibid. 65, 171-178 (1985; Zbl 0584.15009)] on lower bounds on the spectra of symmetric matrices with non-negative entries is proved without the restriction \(\mu\geq 0\) and without requiring the Perron-Frobenius theory. The results by \textit{M. Fiedler} [Czech. Math. J. 25(100), 607-618 (1975; Zbl 0325.15014)] on the above matrices with acyclic graph are generalized.
essentially nonnegative symmetric matrix
scientific article; zbMATH DE number 4093655