Approximately angle preserving mappings

TL;DR

This paper characterizes linear mappings that approximately preserve angles, introducing new concepts and formulas to quantify how closely these mappings maintain angular relationships.

Contribution

It introduces the concept of $(oldsymbol{ ext{ extit{ε}}, c})$-angle preserving mappings and provides an exact formula for their measure based on operator norms.

Findings

Defined $( ext{ extit{ε}}, c)$-angle preserving mappings.

Derived an exact formula for $oldsymbol{ ext{ extit{ε}}}(T, c)$.

Characterized approximately angle preserving mappings.

Abstract

In this paper, we present some characterizations of linear mappings, which preserve vectors at a specific angle. We introduce the concept of $(\varepsilon, c)$-angle preserving mappings for $|c|<1$ and $0\leq \varepsilon < 1 + |c|$. In addition, we define $\widehat{\varepsilon}\,(T, c)$ as the ``smallest'' number $\varepsilon$ for which $T$ is $(\varepsilon, c)$-angle preserving mapping. We state some properties of the function $\widehat{\varepsilon}\,(., c)$, and then propose an exact formula for $\widehat{\varepsilon}\,(T, c)$ in terms of the norm $\|T\|$ and the minimum modulus $[T]$ of $T$. Finally, we characterize the approximately angle preserving mappings.