On n^th Roots of Unity and k^th Order Involutive Matrices

Keleş H., Hall F. J.

WSEAS TRANSACTIONS ON MATHEMATICS, vol.25, pp.202-214, 2026 (Scopus)

Abstract

The nth roots of unity play a central role in algebra and complex analysis, with important applications in number theory, Fourier analysis, and quantum mechanics. This paper investigates the solutions of the equation xn = 1 in both real and complex domains, emphasizing their algebraic structure
and geometric representation.
We further introduce kth order involutive matrices as a generalization of classical involutive matrices and analyze their fundamental spectral properties, including eigenvalue distributions anth roots of unity and the eigenvalues of kth order involutive matrices is established, providing new insight into their cyclic behavior. These results underscore the relevance of higher-order involutive matrices in linear algebra, quantum mechanics, and computational mathematics.