Explicit eigenvalues of certain scaled trigonometric matrices

TL;DR

This paper derives explicit eigenvalues for a class of scaled trigonometric matrices, generalizing previous results and providing proofs for earlier conjectures related to FIR filter design.

Contribution

It introduces a broader class of trigonometric matrices, determines their rank, and provides closed-form eigenvalues, confirming previous conjectures.

Findings

Eigenvalues expressed in closed form

Matrix rank determined as a key property

Provides proof for earlier conjectured eigenvalues

Abstract

In a very recent paper "\emph{On eigenvalues and equivalent transformation of trigonometric matrices}" (D. Zhang, Z. Lin, and Y. Liu, LAA 436, 71--78 (2012)), the authors motivated and discussed a trigonometric matrix that arises in the design of finite impulse response (FIR) digital filters. The eigenvalues of this matrix shed light on the FIR filter design, so obtaining them in closed form was investigated. Zhang \emph{et al.}\ proved that their matrix had rank-4 and they conjectured closed form expressions for its eigenvalues, leaving a rigorous proof as an open problem. This paper studies trigonometric matrices significantly more general than theirs, deduces their rank, and derives closed-forms for their eigenvalues. As a corollary, it yields a short proof of the conjectures in the aforementioned paper.