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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 Generalized singular value decomposition ： ウィキペディア英語版 Generalized singular value decomposition In linear algebra, the generalized singular value decomposition (GSVD) is the name of two different techniques based on the singular value decomposition. The two versions differ because one version decomposes two (or more) matrices (much like higher order PCA) and the other version uses a set of constraints imposed on the left and right singular vectors. ==Higher order version== The generalized singular value decomposition (GSVD) is a matrix decomposition more general than the singular value decomposition. It is used to study the conditioning and regularization of linear systems with respect to quadratic semi-norms. Let $\backslash mathbb\; =\; \backslash mathbb$, or $\backslash mathbb\; =\; \backslash mathbb$. Given matrices $A\; \backslash in\; \backslash mathbb^$ and $B\; \backslash in\; \backslash mathbb^$, their GSVD is given by :$A=U\backslash Sigma\_1\; (X,\; 0)\; Q^$ * and :$B=V\backslash Sigma\_2\; (X,\; 0)\; Q^$ * where $U\; \backslash in\; \backslash mathbb^,V\; \backslash in\; \backslash mathbb^$, and $Q\; \backslash in\; \backslash mathbb^$ are unitary matrices, and $X\; \backslash in\; \backslash mathbb^$ is non-singular, where . Also, $\backslash Sigma\_1\; \backslash in\; \backslash mathbb^$ is non-negative diagonal, and $\backslash Sigma\_2\; \backslash in\; \backslash mathbb^$ is non-negative block-diagonal, with diagonal blocks; $\backslash Sigma\_2$ is not always diagonal. It holds that $\backslash Sigma\_1^T\; \backslash Sigma\_1\; =\; \backslash lceil\backslash alpha\_1^2,\; \backslash dots,\; \backslash alpha\_r^2\backslash rfloor$ and $\backslash Sigma\_2^T\; \backslash Sigma\_2\; =\; \backslash lceil\backslash beta\_1^2,\; \backslash dots,\; \backslash beta\_r^2\backslash rfloor$, and that $\backslash Sigma\_1^T\; \backslash Sigma\_1\; +\; \backslash Sigma\_2^T\; \backslash Sigma\_2\; =\; I\_r$. This implies $0\; \backslash le\; \backslash alpha\_i,\backslash beta\_i\backslash le\; 1$. The ratios $\backslash sigma\_i=\backslash alpha\_i/\backslash beta\_i$ are called the ''generalized singular values'' of $A$ and $B$. If $B$ is square and invertible, then the generalized singular values ''are'' the singular values, and $U$ and $V$ are the matrices of singular vectors, of the matrix $AB^$. Further, if $B\; =\; I$, then the GSVD reduces to the singular value decomposition, explaining the name. 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「Generalized singular value 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.