On explicit formulas of hyperbolic tangent matrix function

The hyperbolic tangent matrix function is a mathematical tool with diverse applications in various fields of mathematics, applied science, and engineering. The hyperbolic tangent matrix th(M), (M is an element of Mr (C)) can be determined through different techniques. In this paper, we establish several explicit formulas for th(tM). The first approach is based on Hermite matrix polynomial expansions, the second utilizes the decomposition of Fibonacci-H & ouml;rner, and the third involves polynomial decomposition utilizing the expression for the n-th power of the matrix and the polynomial decomposition of the exponential matrix. An example is provided to illustrate these methods.