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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 Polar decomposition ： ウィキペディア英語版 Polar decomposition In mathematics, particularly in linear algebra and functional analysis, the polar decomposition of a matrix or linear operator is a factorization analogous to the polar form of a nonzero complex number ''z'' as $z\; =\; r\; e^\backslash ,$ where ''r'' is the absolute value of ''z'' (a positive real number), and $e^$ is an element of the circle group. ==Matrix polar decomposition== The polar decomposition of a square complex matrix ''A'' is a matrix decomposition of the form :$A\; =\; UP\backslash ,$ where ''U'' is a unitary matrix and ''P'' is a positive-semidefinite Hermitian matrix. Intuitively, the polar decomposition separates ''A'' into a component that stretches the space along a set of orthogonal axes, represented by ''P'', and a rotation (with possible reflection) represented by ''U''. The decomposition of the complex conjugate of $A$ is given by $\backslash overline\; =\; \backslash overline$ $\backslash overline$. This decomposition always exists; and so long as ''A'' is invertible, it is unique, with ''P'' positive-definite. Note that :$\backslash det\; A\; =\; \backslash det\; U\backslash ,\backslash det\; P\; =\; re^$ gives the corresponding polar decomposition of the determinant of ''A'', since $\backslash det\; P\; =\; r\; =\; |\backslash det\; A|$ and $\backslash det\; U\; =\; e^$. The matrix ''P'' is always unique, even if ''A'' is singular, and given by :$P\; =\; \backslash sqrt$ where ''A'' * denotes the conjugate transpose of ''A''. This expression is meaningful since a positive-semidefinite Hermitian matrix has a unique positive-semidefinite square root. If ''A'' is invertible, then the matrix ''U'' is given by :$U\; =\; AP^.\backslash ,$ In terms of the singular value decomposition of ''A'', ''A = W Σ V *'', one has :$P\; =\; V\; \backslash Sigma\; V^$ *\, :$U\; =\; W\; V^$ *\, confirming that ''P'' is positive-definite and ''U'' is unitary. Thus, the existence of the SVD is equivalent to the existence of polar decomposition. One can also decompose ''A'' in the form :$A\; =\; P\text{'}U\backslash ,$ Here ''U'' is the same as before and ''P''′ is given by :$P\text{'}\; =\; UPU^\; =\; \backslash sqrt\; =\; W\; \backslash Sigma\; W^$ *. This is known as the left polar decomposition, whereas the previous decomposition is known as the right polar decomposition. Left polar decomposition is also known as reverse polar decomposition. The matrix ''A'' is normal if and only if ''P''′ = ''P''. Then ''UΣ = ΣU'', and it is possible to diagonalise ''U'' with a unitary similarity matrix ''S'' that commutes with ''Σ'', giving ''S U S *'' = ''Φ−1'', where ''Φ'' is a diagonal unitary matrix of phases ''eiφ''. Putting ''Q = V S *'', one can then re-write the polar decomposition as :$A\; =\; (Q\; \backslash Phi\; Q^$ *)(Q \Sigma Q^ *),\, so ''A'' then thus also has a spectral decomposition :$A\; =\; Q\; \backslash Lambda\; Q^$ * \, with complex eigenvalues such that ''ΛΛ * = Σ2'' and a unitary matrix of complex eigenvectors ''Q''. 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「Polar decomposition」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.