Elliptic matrices and their eigenpolynomials
A real symmetric matrix is called elliptic if it has exactly one, simple positive eigenvalue. Elliptic matrices with all diagonal entries equal to zero are called `special' and have been discussed by \textit{M. Fiedler} [Linear Algebra Appl. 197-198, 337-347 (1994; Zbl 0799.15013)]. The author investigates the principal submatrices and the eigenpolynomials of elliptic matrices. As an application a simple proof is given for the following result due to J. H. Smith: A simple graph of order \(n\) is completely \(k\)-partite if, and only if, it is a simply connected graph with elliptic adjacency matrix of rank \(k\).
scientific article; zbMATH DE number 1044244
scientific article