The Intermediate Value Theorem for Linear Transformations

TL;DR

This paper explores a generalization of the Intermediate Value Theorem to linear transformations in higher-dimensional spaces, identifying specific matrix classes that satisfy this property.

Contribution

It characterizes the matrices that satisfy a higher-dimensional analogue of the Intermediate Value Theorem, focusing on monotone and weakly monotone matrices.

Findings

Monotone matrices satisfy a generalized Intermediate Value Theorem.

Weakly monotone matrices also satisfy the theorem under certain conditions.

Applications in numerical solutions of linear systems are discussed.

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.