More about zeros in negative matrices

Frobenius discovered a remarkable link between the spectral properties of an irreducible matrix and its zero pattern, that is, the arrangement of its zero entries (Henryk Minc, in Non-Negative Matrices). This work focuses on studying how the zeros in the powers of a non-negative matrix evolve over time. Let M, (or, occasionally,N or A) denote an nxn matrix. When clarity is needed, we distinguish between the “empty” matrix—which we call a “table”—and the usual matrix, in which numerical values are assigned to its entries.If M is a non-negative matrix, which we will henceforth assume to be n*nn×n with entries in a commutative field, the zeros of M^t are the entries equal to zero. The support of the zeros of M^t is the set of positions where M^t has a zero, denoted Z(M^t) .The Zeros in the powers of M can be classified into several types: ephemeral (disappearing over time), persistent (remaining zero in every power M^t , or periodic (where the pattern of zeros repeats periodically).The behavior of these zeros is closely related to the concepts of reducibility and primitivity of matrices. In particular, persistent zeros tend to form structures we call swarms, which represent obstacles to irreducibility. Accounting for these swarms allows for an elementary proof of Frobenius’s normal form.We also distinguish between the case where the diagonal of M is positive and the case where it is not. This distinction helps in analyzing the potential primitivity of M M as well as the possible periodicity of its powers.