Explicit formulas for computing matrix trigonometric functions

Abstract

Various methods have been introduced for calculating the matrix sine and cosine. This paper concerns the Fibonacci-Horner decomposition of the matrix sine and cosine sin(tA), cos(tA) (A ∈ M (r; ℂ), t ∈ ℝ), which is derived from the combinatorial properties of the generalized Fibonacci sequences in the algebra of square matrices. Finally we derive some formulas for the trigonometric functions of f(A) where f is an homographic function.

Subject Classification: (2010):

• 15A16
• 11B39
• 65F60
• 26D05
• 33B10

Keywords:

• Matrix sine
• Matrix cosine
• Generalized Fibonacci sequence
• Fibonacci-Horner decomposition
• Matrix functions