A method to construct irreducible totally nonnegative matrices with a given Jordan canonical form

Abstract

Let 𝐴 ∈ R𝑛×𝑛 be an irreducible totally nonnegative matrix (ITN), that is, 𝐴 is irreducible with all its minors nonnegative. A triple (𝑛, 𝑟, 𝑝) is called realizable if there exists an ITN matrix 𝐴 ∈ R𝑛×𝑛 with rank(𝐴) = 𝑟 and 𝑝-rank(𝐴) = 𝑝 (recall that 𝑝-rank(𝐴) is the size of the largest invertible principal submatrix of 𝐴). Each ITN matrix 𝐴 associated with a realizable triple (𝑛, 𝑟, 𝑝) has 𝑝 positive and distinct eigenvalues, and for the zero eigenvalue it is verified that 𝑛 − 𝑟 and 𝑛 − 𝑝 are the geometric and the algebraic multiplicity, respectively. Moreover, since rank(𝐴𝑝) = 𝑝, 𝐴 has 𝑛 − 𝑟 zero Jordan blocks whose sizes are given by the Segre characteristic, 𝑆 = (𝑠1, 𝑠2, . . . , 𝑠𝑛−𝑟 ), with 𝑠𝑖 ≤ 𝑝, 𝑖 = 1, 2, . . . , 𝑛 − 𝑟. We know the number of zero Jordan canonical forms of ITN matrices associated with a realizable triple (𝑛, 𝑟, 𝑝) and all these zero Jordan canonical forms. The following important question that we present in this talk deals with how to construct an ITN matrix 𝐴 associated with (𝑛, 𝑟, 𝑝) and exactly with one of these Segre characteristic 𝑆 corresponding to the zero eigenvalue.