Powers of Sign Portraits of Real Matrices

The sign portrait S of a real n× n matrix is a matrix over the semiring with elements 0, 1, -1, and θ, where θ symbolizes indeterminateness. It is proved that if k is the least positive integer such that all the entries of Sk are equal to θ, then k ≤ 2n2 – 3n + 2, and this bound is sharp. Bibliography: 6 titles.