Discrete cosine and sine functions of matrices

Ferhan M. Atıcı; Amber Wu; Ferhan M. Atıcı; Amber Wu

doi:10.3934/era.2026053

Electronic Research Archive

2026, Volume 34, Issue 2: 1157-1172. doi: 10.3934/era.2026053

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Discrete cosine and sine functions of matrices

Ferhan M. Atıcı ^{1
,
,},
Amber Wu ²

1.
Department of Mathematics, Western Kentucky University, Bowling Green, Kentucky 42101, USA
2.
Gatton Academy of Mathematics and Science, Bowling Green, Kentucky 42101, USA

Received: 12 December 2025 Revised: 08 January 2026 Accepted: 13 January 2026 Published: 05 February 2026

In this paper, we introduce two methods for constructing matrix-valued cosine and sine functions defined on a discrete domain. The first method develops an algorithm to compute the $ n \times n $ matrix-valued cosine and sine of a given square matrix $ A $ with some restrictions on its eigenvalues. The second method derives a formula based on the Jordan canonical form to compute each term of a Jordan block in the discrete matrix-valued cosine and sine of a given square matrix $ A $. To illustrate the utility of our approach, we present some examples.
- matrix valued functions,
- Putzer algorithm,
- discrete cosine function,
- Jordan canonical form,
- difference equations
Citation: Ferhan M. Atıcı, Amber Wu. Discrete cosine and sine functions of matrices[J]. Electronic Research Archive, 2026, 34(2): 1157-1172. doi: 10.3934/era.2026053

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Abstract

In this paper, we introduce two methods for constructing matrix-valued cosine and sine functions defined on a discrete domain. The first method develops an algorithm to compute the $ n \times n $ matrix-valued cosine and sine of a given square matrix $ A $ with some restrictions on its eigenvalues. The second method derives a formula based on the Jordan canonical form to compute each term of a Jordan block in the discrete matrix-valued cosine and sine of a given square matrix $ A $. To illustrate the utility of our approach, we present some examples.

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