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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 Block matrix pseudoinverse ： ウィキペディア英語版 Block matrix pseudoinverse In mathematics, block matrix pseudoinverse is a formula of pseudoinverse of a partitioned matrix. This is useful for decomposing or approximating many algorithms updating parameters in signal processing, which are based on least squares method. == Derivation == Consider a column-wise partitioned matrix: :$(A,\; \backslash mathbf\; B),\; \backslash qquad\; \backslash mathbf\; A\; \backslash in\; \backslash reals^,\; \backslash qquad\; \backslash mathbf\; B\; \backslash in\; \backslash reals^,\; \backslash qquad\; m\; \backslash geq\; n+p.$ If the above matrix is full rank, the pseudoinverse matrices of it and its transpose are as follows. :$$ \begin \mathbf A, & \mathbf B \end ^ = ((A, \mathbf B )^T (A, \mathbf B ))^ (A, \mathbf B )^T, :$$ \begin \mathbf A^T \\ \mathbf B^T \end ^ = (A, \mathbf B ) ((A, \mathbf B )^T (A, \mathbf B ))^. The pseudoinverse requires (''n'' + ''p'')-square matrix inversion. To reduce complexity and introduce parallelism, we derive the following decomposed formula. From a block matrix inverse$\backslash mathbf\; ((A,\; \backslash mathbf\; B)^T\; (A,\; \backslash mathbf\; B))^$, we can have :$$ \begin \mathbf A, & \mathbf B \end ^ = \left(P_B^\perp \mathbf A( \mathbf A^T \mathbf P_B^\perp \mathbf A)^, \quad \mathbf P_A^\perp \mathbf B(\mathbf B^T \mathbf P_A^\perp \mathbf B)^\right )^T, :$$ \begin \mathbf A^T \\ \mathbf B^T \end ^ = \left(P_B^\perp \mathbf A( \mathbf A^T \mathbf P_B^\perp \mathbf A)^, \quad \mathbf P_A^\perp \mathbf B(\mathbf B^T \mathbf P_A^\perp \mathbf B)^\right ), where orthogonal projection matrices are defined by ::$$ \begin \mathbf P_A^\perp & = \mathbf I - \mathbf A (\mathbf A^T \mathbf A)^ \mathbf A^T, \\ \mathbf P_B^\perp & = \mathbf I - \mathbf B (\mathbf B^T \mathbf B)^ \mathbf B^T. \end Interestingly, from the idempotence of projection matrix, we can verify that the pseudoinverse of block matrix consists of pseudoinverse of projected matrices: :$$ \begin \mathbf A, & \mathbf B \end ^ = \begin (\mathbf P_B^\mathbf A)^ \\ (\mathbf P_A^\mathbf B)^ \end, :$$ \begin \mathbf A^T \\ \mathbf B^T \end ^ = (A^T \mathbf P_B^)^, \quad (\mathbf B^T \mathbf P_A^)^ ). Thus, we decomposed the block matrix pseudoinverse into two submatrix pseudoinverses, which cost ''n''- and ''p''-square matrix inversions, respectively. Note that the above formulae are not necessarily valid if $(A,\; \backslash mathbf\; B)$ does not have full rank – for example, if $\backslash mathbf\; A\; \backslash neq\; 0$, then :$$ \begin \mathbf A, & \mathbf A \end ^ = \frac \begin \mathbf A^ \\ \mathbf A^ \end \neq \begin (\mathbf P_A^\mathbf A)^ \\ (\mathbf P_A^\mathbf A)^ \end = 0 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「Block matrix pseudoinverse」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.