Abstract: The $$n^{\mathrm{th}}$$ 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 $$x^n=1$$ in both real and complex domains, emphasizing their algebraic structure and geometric representation. We further introduce $$k^{\mathrm{th}}$$-order involutive matrices as a generalization of classical involutive matrices and analyze their fundamental spectral properties, including eigenvalue distributions. A relationship between the $$n^{\mathrm{th}}$$ roots of unity and the eigenvalues of $$k^{\mathrm{th}}$$-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.

Keywords: Roots of unity, involutive matrices, $$k^{\mathrm{th}}$$-order involutive, eigenvalues, linear algebra, spectral analysis, Fourier analysis, quantum mechanics.

Pages: 202-214