Euler-Rodrigues formula for three-dimensional rotation via fractional powers of matrices

TL;DR

This paper reviews the Euler-Rodrigues formula for 3D rotations and introduces a method to derive rotations at any angle using fractional powers of the rotation matrix, based on spectral analysis.

Contribution

It presents a novel approach to obtain arbitrary rotations in 3D by fractional powers of the rotation matrix, expanding the understanding of matrix spectral behavior.

Findings

Derivation of 3D rotations using fractional powers of matrices.

Spectral analysis of rotation matrices for arbitrary angles.

Extension of Euler-Rodrigues formula via fractional calculus.

Abstract

In this short paper, we review the Euler-Rodrigues formula for three-dimensional rotation via fractional powers of matrices. We derive the rotations by any angle through the spectral behavior of the fractional powers of the rotation matrix by $\frac{\pi}{2}$ in $\mathbb{R}^3$ about some axis.