Generalization and Alternative Proof of Two Identities Posed by Sun

TL;DR

This work studies two Sun identities linking derangements, roots of unity, and determinants of Hermitian matrices. It generalizes these identities to the circulant-matrix setting, establishing a determinant formula for $(n-1)\times(n-1)$ minors of circulant matrices $M=(f(i-j))$ with eigenvalues $\lambda_i(M)=\sum_{k=0}^{n-1} f(k)\zeta^{-ki}$ and $\det M_j=\frac{1}{n}\prod_{i\neq s}\lambda_i(M)$ when a discrete Fourier coefficient vanishes. An independent EVT-free proof is then provided by constructing a Hermitian matrix with off-diagonal entries $a_{ij}=\frac{1}{1-x_{i-j}}$ (with $x_k=\zeta^k$) and analyzing $C_s=AB_s$ via a Fourier-based eigenbasis to obtain integer eigenvalues that reproduce Sun's identities, including special prime-$n$ cases and generalizations with $f(k)=(a+b\zeta^{ck})/(1-\zeta^k)$. Collectively, the results deepen the link between derangements, roots of unity, and circulant-determinant structures, offering new techniques for proving trigonometric-determinant identities and suggesting avenues for further generalizations.

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.

Theorems & Definitions (7)

• Theorem 1.1
• Theorem 1.2
• Theorem 2.1
• proof
• Theorem 2.2
• Lemma 4.1
• proof