Conformal Mapping among Orthogonal, Symmetric, and Skew-Symmetric Matrices
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This paper shows that Cayley Transforms, which map Orthogonal and Skew-Symmetric matrices, may be considered the extension to matrix field of the complex conformal mapping function f 1(z) = 1-z/1+x. Then, by using a set of real matrices which are, simultaneously, Orthogonal and Symmetric (the Ortho-Sym matrices), it similarly shows how to extend two complex conformal mapping functions (namely, the f 2(z) = i-z/i+z, and the f 3(z) = 1-z/1+z i - here called the clockwise, and the counter clockwise functions), to matrix field. This extension consists of some new one-to-one mapping relationships between Orthogonal and Symmetric, and between Symmetric and Skew-Symmetric matrices. This new relationships complete the picture of the one-to-one matrix mapping among Orthogonal, Symmetric, and Skew-Symmetric matrices. Finally, this paper shows how to map among Orthogonal, Symmetric, and Skew-Symmetric matrices, by means of a direct product.