Generalization and Alternative Proof of Two Identities Posed by Sun

TL;DR

This paper generalizes two identities involving roots of unity and Hermitian matrices to circulant matrices, providing an alternative proof that does not rely on the eigenvector-eigenvalue identity, thus broadening their applicability.

Contribution

It extends Sun's identities to circulant matrices and offers a new proof method based on matrix similarity and Fourier transform vectors.

Findings

Identities are generalized to circulant matrices.

An alternative proof method is developed independent of eigenvector-eigenvalue identity.

The approach utilizes matrix similarity and Fourier transform vectors.

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.