Given a totally positive matrix, can one insert a line (row or column) between two given lines while maintaining total positivity? This question was first posed and solved by Johnson and Smith who gave an algorithm that results in one possible line insertion. In this work, we revisit this problem. First, we show that every totally positive matrix can be associated with a certain vertex-weighted graph in such a way that the entries of the matrix are equal to sums over certain paths in this graph. We call this graph a scaffolding of the matrix. We then use this to give a complete characterization of all possible line insertions as the strongly positive solutions to a given homogeneous system of linear equations.

给定一个完全正的矩阵，可以在两条给定线之间插入一行（行或列），同时保持总正性吗？这个问题首先由约翰逊和史密斯提出并解决，他们给出了一种算法，可以产生一个可能的行插入。在这项工作中，我们重新审视这个问题。首先，我们证明每个全正矩阵都可以与某个顶点加权图相关联，使得矩阵的条目等于该图中某些路径上的总和。我们将此图称为矩阵的脚手架。然后，我们用它来完整描述所有可能的直线插入，作为给定齐次线性方程组的强正解。