AI 中文总结
本文解决了半正矩阵可逆线性保持映射猜想中m<n的情形，其证明适用于任意正整数m、n，核心是对半正矩阵集合中最大维度仿射子空间分类。
AI 中文摘要
若m×n实矩阵A存在向量x>0使得Ax>0（不等式按分量理解），则称A为半正矩阵。Dorsey等人猜想，任何保持所有半正矩阵集合不变的可逆线性映射L，始终具有标准形式A↦XAY，其中X为行正矩阵，Y为非负逆矩阵。当m≥n时该猜想已获解决，本文旨在解决m<n的情形，且证明适用于任意正整数m、n，核心是对半正矩阵集合中最大可能维度的仿射子空间进行分类。
英文摘要
An m-by-n real matrix A is said to be semipositive if there exists a vector x>0 such that Ax>0, where the inequalities are understood componentwise. Dorsey et al. conjectured that any invertible linear map that $L$ that leaves invariant the collection of all semipositive matrices is always in the standard form $A \mapsto XAY$ for some row positive matrix $X$ and inverse nonnegative matrix $Y$. This was settled in when $m \geq n$. Our aim in this paper is to settle the case when m<n of the above conjecture. In fact, our proof works for arbitrary positive integers m and n. The main ingredient is a classification of affine subspaces of the largest possible dimension contained in the set of semipositive matrices.
