An inclusion region for the field of values of a doubly stochastic matrix based on its graph

TLDR: In this article, it was shown that if A = (%) is an n-by-n entry-wise nonnegative matrix whose Perron-Frobenius eigenvalue is 1, then o(A) c Ln (1) where o-(A) denotes the spectrum or set of all eigenvalues of A. Since the unit circle U is not contained in L, this yields an improvement on the usual inclusion region, namely: ~ r (A)c U, for the eigen values of A (which may be obtained via Gersgorin

Abstract: of the complex plane by Lk. This is precisely the region in which complex numbers u + iv satisfy u + Ivl tan (w/k)-< 1. It has been shown [1, 5, 7] that if A = (%) is an n-by-n entry-wise nonnegative matrix whose Perron-Frobenius eigenvalue is 1, then o-(A) c Ln (1) where o-(A) denotes the spectrum or set of all eigenvalues of A. Since the unit circle U is not contained in L,, this yields an improvement on the usual inclusion region, ~ r (A)c U, for the eigenvalues of A (which may be obtained via Gersgorin, norms or Perron-Frobenius), namely: ~r(A) c L~ 71 U. (2)