Properties of The Set of Orthogonal Matrices Obtained as The Cayley Transform of Skew-Symmetric Matrices

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  • 歪対称行列のCayley変換により生成される直交行列集合の性質

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For optimization problems with the orthogonality constraint imposed on matrix variables, many studies use the Cayley transform of skew-symmetric matrices to generate orthogonal matrices, however, such transform has the problem that it cannot entirely generate all orthogonal matrices. The main issue of this paper is to show properties of the image of that transform as a subset of the entire orthogonal matrix. We give theoretical analysis for this point and consequently prove that that transform can always generate the representative of the equivalence class consist of orthonormal bases. This clarifies the range of optimization problems to adopt that transform.

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