Explicit eigenvalues of certain scaled trigonometric matrices

Details Explicit eigenvalues of certain scaled trigonometric matrices Explicit eigenvalues of certain scaled trigonometric matrices

2012-01-23

arxiv.org/abs/1201.4651v1

Mathematics

Numerical Analysis

6 pages

Scientific paper

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 discussed a certain trigonometric matrix that arises in the design of finite impulse response (FIR) digital filters. They conjectured that their matrix had rank 4 and that it admitted a simple closed form for its eigenvalues. We introduce more general trigonometric matrices, for which we present an elementary approach to deduce rank, and to derive eigenvalues in closed-form. As a corollary, we obtain a short proof of the conjecture in the aforementioned paper.

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