Explicit eigenvalues of certain scaled trigonometric matrices

In a recent paper [D. Zhang, Z. Lin, Y. Liu, On eigenvalues and equivalent transformation of trigonometric matrices, Linear Algebra Appl. 436 (2012) 71-78], 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 et al. proved that their matrix is rank-4 and they conjectured closed form expressions for its eigenvalues. This paper studies trigonometric matrices more general than theirs, deduces their rank, and derives closed-forms for their eigenvalues. As a corollary, it yields an elementary proof of the conjecture in the aforementioned paper.