The structure of the set ϐ(A)of all eigenvalues of all complex matrices (elementwise) equimodular with a given n x n non-negative matrix A is studied.The problem was suggested by O. Taussky and some aspects have been studied by R. S. Varga and B.W. Levinger.
If every matrix equimodular with A is non-singular, then A is called regular.A new proof of the P. Camion-A.J. Hoffman characterization of regular matrices is given.
The set ϐ(A)consists of m ≤ n closed annuli centered at the origin.Each gap, ɤ, in this set can be associated with a class of regular matrices with a (unique) permutation, π(ɤ).The association depends on both the combinatorial structure of A and the size of the aii.Let A be associated with the set of r permutations, π1, π2,…, πr, where each gap in ϐ(A) is associated with one of the πk.Then r ≤ n, even when the complement of ϐ(A) has n+1 components.Further, if π(ɤ) is the identity, the real boundary points of ɤ are eigenvalues of real matrices equimodular with A.In particular, if A is essentially diagonally dominant, every real boundary point of ϐ(A) is an eigenvalues of a real matrix equimodular with A.
Several conjectures based on these results are made which if verified would constitute an extension of the Perron-Frobenius Theorem, and an algebraic method is introduced which unites the study of regular matrices with that of ϐ(A).
