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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Cayley transform ： ウィキペディア英語版 Cayley transform In mathematics, the Cayley transform, named after Arthur Cayley, is any of a cluster of related things. As originally described by , the Cayley transform is a mapping between skew-symmetric matrices and special orthogonal matrices. In complex analysis, the Cayley transform is a conformal mapping in which the image of the upper complex half-plane is the unit disk . And in the theory of Hilbert spaces, the Cayley transform is a mapping between linear operators . == Matrix map == Among ''n''×''n'' square matrices over the reals, with ''I'' the identity matrix, let ''A'' be any skew-symmetric matrix (so that ''A''T = −''A''). Then ''I'' + ''A'' is invertible, and the Cayley transform :$Q\; =\; (I\; -\; A)(I\; +\; A)^\; \backslash ,\backslash !$ produces an orthogonal matrix, ''Q'' (so that ''Q''T''Q'' = ''I''). The matrix multiplication in the definition of ''Q'' above is commutative, so ''Q'' can be alternatively defined as $Q\; =\; (I\; +\; A)^(I\; -\; A)$. In fact, ''Q'' must have determinant +1, so is special orthogonal. Conversely, let ''Q'' be any orthogonal matrix which does not have −1 as an eigenvalue; then :$A\; =\; (I\; -\; Q)(I\; +\; Q)^\; \backslash ,\backslash !$ is a skew-symmetric matrix. The condition on ''Q'' automatically excludes matrices with determinant −1, but also excludes certain special orthogonal matrices. Some authors use a superscript "c" to denote this transform, writing ''Q'' = ''A''c and ''A'' = ''Q''c. This version of the Cayley transform is its own functional inverse, so that ''A'' = (''A''c)c and ''Q'' = (''Q''c)c. A slightly different form is also seen , requiring different mappings in each direction (and dropping the superscript notation): :$\backslash begin$ Q &= (Q - I)(Q + I)^ \end The mappings may also be written with the order of the factors reversed ; however, ''A'' always commutes with (μ''I'' ± ''A'')−1, so the reordering does not affect the definition. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Cayley transform」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.