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

Abstract

[EN] 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 associated with a realizable triple (n,r,p) has 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(A)=r, has n¿r zero Jordan blocks whose sizes are given by the Segre characteristic, S= (s1, s2,.. . , s_n-r ), with s_i¿ 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 associated with (n,r,p) and exactly with one of these Segre characteristic S corresponding to the zero eigenvalue.