The Intermediate Value Theorem for Linear Transformations
Ruben A. Martinez-Avendaño
TL;DR
The paper addresses a higher-dimensional analogue of the Intermediate Value Theorem for linear maps between Euclidean spaces endowed with a coordinatewise partial order. It demonstrates that naive generalizations fail and introduces weakly monotone (and monotone) matrices as the right objects to capture an IVT-like property in this setting. Using tools from convex analysis and Farkas' Lemma, it establishes equivalences of preimage-interval properties, derives sufficient conditions via nonnegative left inverses, and discusses rank considerations, providing several corollaries and geometric interpretations. The results have implications for numerical methods solving $A\mathbf{x}=\mathbf{b}$ by justifying the search for intermediate solutions within domain bounds, while also outlining open questions and potential nonlinear extensions related to classical theorems like Poincaré-Miranda.
Abstract
If a real-valued function is continuous on a real interval and it takes on two different values, then it will also take any value in between those two, by the Intermediate Value Theorem. It is not immediately clear what would be a natural generalization for functions whose domain and range are in higher-dimensional Euclidean spaces. In this article, we analyze this problem, by first arriving at what we think is the appropriate question to ask, and then restricting to linear transformations. It turns out that the matrices that will satisfy an appropriate version of the Intermediate Value Theorem are the so called {\em monotone} and {\em weakly monotone} matrices, which have applications in numerical approximation of the solutions to systems of linear equations.