On the Spectral Region of 4-Cycle Stochastic Matrices

We study the spectrum of 4-cycle row-stochastic matrices. For real eigenvalues the spectral region is [-1,1]. For nonreal eigenvalues a+ib we derive necessary conditions in terms of the real and imaginary parts, including the inequality a+|b| <= 1 and the condition (b^2+a^2+a)^2+2a^2-b^2 >= 0. We also prove conversely that every point in the corresponding interior region occurs as an eigenvalue of a 4-cycle matrix. The proof is organized through a reformulation of the characteristic equation, an argument parametrization, a convex-analytic criterion, and explicit boundary constructions. Hence, the spectral region for the 4-cycle row-stochastic matrices is exactly and explicitly determined.