Ortho-skew and ortho-sym matrix trigonometry
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This paper introduces some properties of two sets of matrices, the Ortho-Skew, which are simultaneously Orthogonal and Skew-Hermitian, and the real Ortho-Sym matrices, which are Orthogonal and Symmetric. The introduced relationships, all demonstrated, consist of closed-form compact expressions of trigonometric and hyperbolic functions that show these matrices working as angles in the matrix field. The analogies with trigonometric and hyperbolic functions, such as the periodicity of the trigonometric functions and some properties of the inverse functions are all shown. Additional expressions are derived for 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 considered as the extension to the matrix field of the imaginary and the real units, respectively.
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