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This is a Quantitative Study study.

October 8, 2025

Generalization and alternative proof of two identities posed by Sun

Key Points

The study generalizes two identities involving roots of unity, extending their applicability through circulant matrices.
Key findings reveal alternative proofs of Sun's identities, independent of the eigenvector-eigenvalue identity used previously.
The research employs mathematical techniques linking normal matrices to ones with integer eigenvalues through the Fourier transform.
These findings signify potential new approaches for handling similar identities in matrix theory.

Abstract

We study two identities involving roots of unity and determinants of Hermitian matrices which have been recently proved by using the famous eigenvector-eigenvalue identity for normal matrices. In this paper, we extend these identities to a more general form by considering the class of circulant matrices. Furthermore, we give an alternative proof of Sun's identities independent of the eigenvector-eigenvalue identity, where our strategy is built upon the similarity of an unnecessarily normal matrix to a particular matrix with integer eigenvalues, derived from the Fourier transform vectors.

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