On the eigenvectors belonging to the minimum eigenvalue of an essentially nonnegative symmetric matrix with bipartite graph
- Roth, Robert
DOIAbstract
AbstractWe prove that if B is an essentially nonnegative symmetric matrix with minimum eigenvalue m(B) and if the graph of B is connected and bipartite (with bipartition V1 ∪ V2, then there exists an eigenvector x belonging to m(B) such that xi > 0 for iϵV1 and xi<0 for iϵV2. We then establish two facts concerning essentially nonnegative symmetric matrices with bipartite graph and point out how the above result can be used in conjunction with these facts to give a proof, not requiring the Perron-Frobenius theory, of the following theorem of Constantine [2]: Among all essentially nonnegative symmetric n-by-n matrices with minimum diagonal entry at least μ and maximum off-diagonal entry at most M, the matrix with smallest minimum eigenvalue i...
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