Ortho-Skew and Ortho-Sym matrix trigonometry

This paper introduces some properties of two families of matrices: the Ortho-Skew, which are simultaneously orthogonal and skew-Hermitian, and the real Ortho-Sym matrices, which are orthogonal and symmetric. These relationships consist of closed-form compact expressions of trigonometric and hyperbolic functions that show that multiples of these matrices can be interpreted as angles. The analogies with trigonometric and hyperbolic functions, such as the periodicity of the trigonometric functions, are all shown. Additional expressions are derived from some other functions of matrices such as the logarithm, exponential, inverse, and power functions. All these relationships show that the Ortho-Skew and the Ortho-Sym matrices can be respectively considered as matrix extensions of the imaginary and the real units.