Let \(A \in {\mathbb {R}}^{n \times n}\) be a totally nonpositive matrix (t.n.p.) with rank r and principal rank p, that is, every minor of A is nonpositive and p is the size of the largest invertible principal submatrix of A. We introduce that a triple (n, r, p) will be called negatively realizable if there exists a t.n.p. matrix A of order n and such that its rank is r and its principal rank is p. In this work we extend the results obtained for irreducible totally nonnegative matrices given in Cantó and Urbano (Linear Algebra Appl 551:125–146. https://doi.org/10.1016/j.laa.2018.03.045, 2018) to t.n.p. matrices. For that, we consider the sequence of the first p-indices of A and study the linear dependence relations between their rows and columns. These relations allow us to construct t.n.p. matrices associated with a triple (n, r, p) negatively realizable and a specific sequence of the first p-indices.

令\(A \in {\mathbb {R}}^{n \times n}\)是秩为r且主秩为p的完全非正矩阵 (tnp) ，即A 的每个次要矩阵都是非正的，而p是A的最大可逆主子矩阵的大小。我们引入一个三元组 ( n ,  r ,  p ) 如果存在一个n阶的 tnp 矩阵A并且它的秩为r并且它的主秩为p ，则称其为负可实现. 在这项工作中，我们将 Cantó 和 Urbano (Linear Algebra Appl 551:125–146. https://doi.org/10.1016/j.laa.2018.03.045, 2018) 中给出的不可约完全非负矩阵的结果扩展到 tnp矩阵。为此，我们考虑A的第一个p索引的序列并研究它们的行和列之间的线性相关关系。这些关系使我们能够构建与三元组 ( n ,  r ,  p ) 负可实现相关的 tnp 矩阵和第一个p索引的特定序列。