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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 Gram–Schmidt process ： ウィキペディア英語版 Gram–Schmidt process In mathematics, particularly linear algebra and numerical analysis, the Gram–Schmidt process is a method for orthonormalising a set of vectors in an inner product space, most commonly the Euclidean space R''n''. The Gram–Schmidt process takes a finite, linearly independent set ''S'' = for and generates an orthogonal set that spans the same ''k''-dimensional subspace of R''n'' as ''S''. The method is named after Jørgen Pedersen Gram and Erhard Schmidt but it appeared earlier in the work of Laplace and Cauchy. In the theory of Lie group decompositions it is generalized by the Iwasawa decomposition. The application of the Gram–Schmidt process to the column vectors of a full column rank matrix yields the QR decomposition (it is decomposed into an orthogonal and a triangular matrix). == The Gram–Schmidt process == We define the projection operator by :$\backslash mathrm\_)\; =\; \backslash rangle\backslash over\backslash langle\; \backslash mathbf,\; \backslash mathbf\backslash rangle\}\backslash mathbf\; ,$ where $\backslash langle\; \backslash mathbf,\; \backslash mathbf\backslash rangle$ denotes the inner product of the vectors v and u. This operator projects the vector v orthogonally onto the line spanned by vector u. If u=0, we define $\backslash mathrm\_0\backslash ,(\backslash mathbf)\; :=\; 0$. i.e., the projection map $\backslash mathrm\_0$ is the zero map, sending every vector to the zero vector. The Gram–Schmidt process then works as follows: : $$ \begin \mathbf_1 & = \mathbf_1, & \mathbf_1 & = _1\|} \\ \mathbf_2 & = \mathbf_2-\mathrm_\,(\mathbf_2), & \mathbf_2 & = _2\|} \\ \mathbf_3 & = \mathbf_3-\mathrm_\,(\mathbf_3)-\mathrm_\,(\mathbf_3), & \mathbf_3 & = _3\|} \\ \mathbf_4 & = \mathbf_4-\mathrm_\,(\mathbf_4)-\mathrm_\,(\mathbf_4)-\mathrm_\,(\mathbf_4), & \mathbf_4 & = _4\|} \\ & _k & = \mathbf_k-\sum_^\mathrm_\,(\mathbf_k), & \mathbf_k & = _k \|}. \end The sequence u1, ..., u''k'' is the required system of orthogonal vectors, and the normalized vectors e1, ..., e''k'' form an ortho''normal'' set. The calculation of the sequence u1, ..., u''k'' is known as ''Gram–Schmidt orthogonalization'', while the calculation of the sequence e1, ..., e''k'' is known as ''Gram–Schmidt orthonormalization'' as the vectors are normalized. To check that these formulas yield an orthogonal sequence, first compute ‹ u1,u2 › by substituting the above formula for u2: we get zero. Then use this to compute ‹ u1,u3 › again by substituting the formula for u3: we get zero. The general proof proceeds by mathematical induction. Geometrically, this method proceeds as follows: to compute u''i'', it projects v''i'' orthogonally onto the subspace ''U'' generated by u1, ..., u''i''−1, which is the same as the subspace generated by v1, ..., v''i''−1. The vector u''i'' is then defined to be the difference between v''i'' and this projection, guaranteed to be orthogonal to all of the vectors in the subspace ''U''. The Gram–Schmidt process also applies to a linearly independent countably infinite sequence ''i''. The result is an orthogonal (or orthonormal) sequence ''i'' such that for natural number ''n'': the algebraic span of v1, ..., v''n'' is the same as that of u1, ..., u''n''. If the Gram–Schmidt process is applied to a linearly dependent sequence, it outputs the 0 vector on the ''i''th step, assuming that v''i'' is a linear combination of . If an orthonormal basis is to be produced, then the algorithm should test for zero vectors in the output and discard them because no multiple of a zero vector can have a length of 1. The number of vectors output by the algorithm will then be the dimension of the space spanned by the original inputs. A variant of the Gram–Schmidt process using transfinite recursion applied to a (possibly uncountably) infinite sequence of vectors $(v\_\backslash alpha)\_$ yields a set of orthonormal vectors $(u\_\backslash alpha)\_$ with $\backslash kappa\backslash leq\backslash lambda$ such that for any $\backslash alpha\backslash leq\backslash lambda$, the completion of the span of is the same as that of . In particular, when applied to a (algebraic) basis of a Hilbert space (or, more generally, a basis of any dense subspace), it yields a (functional-analytic) orthonormal basis. Note that in the general case often the strict inequality holds, even if the starting set was linearly independent, and the span of $(u\_\backslash alpha)\_$ need not be a subspace of the span of $(v\_\backslash alpha)\_$ (rather, it's a subspace of its completion). 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「Gram–Schmidt process」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.