. . . "In linear algebra, orthogonalization is the process of finding a set of orthogonal vectors that span a particular subspace. Formally, starting with a linearly independent set of vectors {v1, ... , vk} in an inner product space (most commonly the Euclidean space Rn), orthogonalization results in a set of orthogonal vectors {u1, ... , uk} that generate the same subspace as the vectors v1, ... , vk. Every vector in the new set is orthogonal to every other vector in the new set; and the new set and the old set have the same linear span. In addition, if we want the resulting vectors to all be unit vectors, then we normalize each vector and the procedure is called orthonormalization. Orthogonalization is also possible with respect to any symmetric bilinear form (not necessarily an inner product, not necessarily over real numbers), but standard algorithms may encounter division by zero in this more general setting."@en . "201611"^^ . "\u76F4\u4EA4\u5316\uFF08\u3061\u3087\u3063\u3053\u3046\u304B\uFF09\u3068\u306F\u3001\u7DDA\u578B\u7A7A\u9593\u4E0A\u306B\u3042\u308B\u30D9\u30AF\u30C8\u30EB\u306E\u7D44\u304B\u3089\u3001\u4E92\u3044\u306B\u76F4\u4EA4\u3059\u308B\u30D9\u30AF\u30C8\u30EB\u306E\u7D44\u3092\u751F\u6210\u3059\u308B\u3053\u3068\u3067\u3042\u308B\u3002"@ja . . "Em \u00E1lgebra linear, ortogonaliza\u00E7\u00E3o \u00E9 o processo de encontrar um conjunto de vetor ortogonal que gera um subespa\u00E7o espec\u00EDfico. Formalmente, come\u00E7ando com um conjunto linearmente independente de vetores {v1,\u202F...\u202F,\u202Fvk} em um espa\u00E7o com produto interno (mais frequentemente o espa\u00E7o euclidiano Rn), o processo de ortogonaliza\u00E7\u00E3o resulta em um conjunto de vetores ortogonais {u1,\u202F...\u202F,\u202Fuk} que geram o mesmo subespa\u00E7o que os vetores v1,\u202F...\u202F,\u202Fvk. Todo vetor do novo conjunto \u00E9 ortogonal a todos os demais vetores do novo conjunto; e o novo conjunto e o antigo possuem o mesmo espa\u00E7o gerado. Al\u00E9m disso, se o objetivo for obter vetores que s\u00E3o unit\u00E1rios, ent\u00E3o o procedimento \u00E9 chamado de ortonormaliza\u00E7\u00E3o. Tamb\u00E9m \u00E9 poss\u00EDvel realizar o processo de ortonormaliza\u00E7\u00E3o com rela\u00E7\u00E3o a qualquer forma bilinear sim\u00E9trica (n\u00E3o necessariamente um produto interno, e n\u00E3o necessariamente sobre os n\u00FAmeros reais), mas os algoritmos usuais podem se deparar com divis\u00E3o por zero nesta situa\u00E7\u00E3o mais geral."@pt . "3595"^^ . "Orthogonalisierungsverfahren"@de . . . . . "1044808212"^^ . . . . . "\u6B63\u4EA4\u5316"@zh . . . "\u041E\u0440\u0442\u043E\u0433\u043E\u043D\u0430\u043B\u0438\u0437\u0430\u0446\u0438\u044F"@ru . "\u041E\u0440\u0442\u043E\u0433\u043E\u043D\u0430\u043B\u0438\u0437\u0430\u0446\u0438\u044F \u2015 \u043F\u0440\u043E\u0446\u0435\u0441\u0441 \u043F\u043E\u0441\u0442\u0440\u043E\u0435\u043D\u0438\u044F \u043F\u043E \u0437\u0430\u0434\u0430\u043D\u043D\u043E\u043C\u0443 \u0431\u0430\u0437\u0438\u0441\u0443 \u043B\u0438\u043D\u0435\u0439\u043D\u043E\u0433\u043E \u043F\u0440\u043E\u0441\u0442\u0440\u0430\u043D\u0441\u0442\u0432\u0430 \u043D\u0435\u043A\u043E\u0442\u043E\u0440\u043E\u0433\u043E \u043E\u0440\u0442\u043E\u0433\u043E\u043D\u0430\u043B\u044C\u043D\u043E\u0433\u043E \u0431\u0430\u0437\u0438\u0441\u0430, \u043A\u043E\u0442\u043E\u0440\u044B\u0439 \u0438\u043C\u0435\u0435\u0442 \u0442\u0443 \u0436\u0435 \u0441\u0430\u043C\u0443\u044E \u043B\u0438\u043D\u0435\u0439\u043D\u0443\u044E \u043E\u0431\u043E\u043B\u043E\u0447\u043A\u0443. \u0412\u0432\u0438\u0434\u0443 \u0443\u0434\u043E\u0431\u0441\u0442\u0432\u0430 \u0438 \u0432\u0430\u0436\u043D\u043E\u0441\u0442\u0438 \u043E\u0440\u0442\u043E\u0433\u043E\u043D\u0430\u043B\u044C\u043D\u044B\u0445 \u0431\u0430\u0437\u0438\u0441\u043E\u0432 \u0432 \u0440\u0430\u0437\u043B\u0438\u0447\u043D\u044B\u0445 \u0437\u0430\u0434\u0430\u0447\u0430\u0445, \u0432\u0430\u0436\u043D\u044B \u0438 \u043F\u0440\u043E\u0446\u0435\u0441\u0441\u044B \u043E\u0440\u0442\u043E\u0433\u043E\u043D\u0430\u043B\u0438\u0437\u0430\u0446\u0438\u0438."@ru . . . . "\u7EBF\u6027\u4EE3\u6570\u4E2D\u7684\u6B63\u4EA4\u5316\u6307\u7684\u662F\uFF1A\u4ECE\u5185\u79EF\u7A7A\u95F4\uFF08\u5305\u62EC\u5E38\u89C1\u7684\u6B27\u51E0\u91CC\u5F97\u7A7A\u95F4\uFF09\u4E2D\u7684\u4E00\u7EC4\u7EBF\u6027\u65E0\u5173\u5411\u91CFv1\uFF0C...\uFF0Cvk\u51FA\u53D1\uFF0C\u5F97\u5230\u540C\u4E00\u4E2A\u5B50\u7A7A\u95F4\u4E0A\u4E24\u4E24\u6B63\u4EA4\u7684\u5411\u91CF\u7EC4u1\uFF0C...\uFF0Cuk\u3002 \u5982\u679C\u8FD8\u8981\u6C42\u6B63\u4EA4\u5316\u540E\u7684\u5411\u91CF\u90FD\u662F\u5355\u4F4D\u5411\u91CF\uFF0C\u90A3\u4E48\u79F0\u4E3A\u6807\u51C6\u6B63\u4EA4\u5316\u3002 \u4E00\u822C\u5728\u6570\u5B66\u5206\u6790\u4E2D\u91C7\u7528\u683C\u62C9\u59C6-\u65BD\u5BC6\u7279\u6B63\u4EA4\u5316\u4F5C\u6B63\u4EA4\u5316\u7684\u8BA1\u7B97\u3002\u5728\u7F16\u7A0B\u8BA1\u7B97\u65F6\uFF0C\u683C\u62C9\u59C6-\u65BD\u5BC6\u7279\u6B63\u4EA4\u5316\u7684\u6570\u503C\u7A33\u5B9A\u6027\u4E0D\u9AD8\uFF0C\u6240\u4EE5\u5E38\u7528\u66F4\u7A33\u5B9A\u7684\u8C6A\u65AF\u970D\u5C14\u5FB7\u53D8\u6362\u4EE3\u66FF\u3002\u53E6\u5916\uFF0C\u76F8\u5BF9\u4E8E\u8C6A\u65AF\u970D\u5C14\u5FB7\u53D8\u6362\u5728\u6700\u540E\u76F4\u63A5\u751F\u6210\u6240\u6709\u7684\u5411\u91CF\uFF0C\u683C\u62C9\u59C6-\u65BD\u5BC6\u7279\u65B9\u6CD5\u5728\u7B2Ci\u6B65\u4EA7\u751F\u7B2Ci\u4E2A\u5411\u91CF\uFF0C\u56E0\u6B64\u540E\u8005\u53EF\u7528\u8FED\u4EE3\u6CD5\u7F16\u5199\u3002\u5BF9\u4E8E\u542B\u6709\u96F6\u5143\u7D20\u8F83\u591A\u7684\u5411\u91CF\u7EC4\uFF08\u4F8B\u5982\u7A00\u758F\u77E9\u9635\u7684QR\u5206\u89E3\uFF09\uFF0C\u8FD8\u4F1A\u91C7\u7528\u5409\u6587\u65AF\u65CB\u8F6C\u3002"@zh . . . "Orthogonalization"@en . . . "\u76F4\u4EA4\u5316"@ja . . . . "Mit Orthogonalisierungsverfahren bezeichnet man in der Mathematik Algorithmen, die aus einem System linear unabh\u00E4ngiger Vektoren ein Orthogonalsystem erzeugen, das den gleichen Untervektorraum aufspannt. Das bekannteste Verfahren dieser Art ist das Gram-Schmidtsche Orthogonalisierungsverfahren. Dieses kann man f\u00FCr beliebige Vektoren aus einem Pr\u00E4hilbertraum verwenden. Oftmals ist die Orthogonalisierung von Vektoren zwar namensgebend, aber nicht das eigentliche Ziel solcher Verfahren. So benutzt man in der Numerischen Mathematik Orthogonalisierungsverfahren wie die Householder-Transformation oder die Givens-Rotation haupts\u00E4chlich um eine QR-Zerlegung mit einer orthogonalen Matrix und einer Dreiecksmatrix zu erzeugen. Die Spaltenvektoren der Matrix sind dann die orthogonalisierten Spaltenvektoren der Matrix . Haupts\u00E4chlich erh\u00E4lt man aber eine stabile Methode zum L\u00F6sen linearer Gleichungssysteme. Zur R\u00FCckf\u00FChrung eines verallgemeinerten Eigenwertproblems auf ein spezielles Eigenwertproblem kann man Symmetrische Orthogonalisierung sowie verwenden."@de . . "Em \u00E1lgebra linear, ortogonaliza\u00E7\u00E3o \u00E9 o processo de encontrar um conjunto de vetor ortogonal que gera um subespa\u00E7o espec\u00EDfico. Formalmente, come\u00E7ando com um conjunto linearmente independente de vetores {v1,\u202F...\u202F,\u202Fvk} em um espa\u00E7o com produto interno (mais frequentemente o espa\u00E7o euclidiano Rn), o processo de ortogonaliza\u00E7\u00E3o resulta em um conjunto de vetores ortogonais {u1,\u202F...\u202F,\u202Fuk} que geram o mesmo subespa\u00E7o que os vetores v1,\u202F...\u202F,\u202Fvk. Todo vetor do novo conjunto \u00E9 ortogonal a todos os demais vetores do novo conjunto; e o novo conjunto e o antigo possuem o mesmo espa\u00E7o gerado."@pt . . . . . . . . . "Ortogonaliza\u00E7\u00E3o"@pt . . "In linear algebra, orthogonalization is the process of finding a set of orthogonal vectors that span a particular subspace. Formally, starting with a linearly independent set of vectors {v1, ... , vk} in an inner product space (most commonly the Euclidean space Rn), orthogonalization results in a set of orthogonal vectors {u1, ... , uk} that generate the same subspace as the vectors v1, ... , vk. Every vector in the new set is orthogonal to every other vector in the new set; and the new set and the old set have the same linear span."@en . . . . . "\u7EBF\u6027\u4EE3\u6570\u4E2D\u7684\u6B63\u4EA4\u5316\u6307\u7684\u662F\uFF1A\u4ECE\u5185\u79EF\u7A7A\u95F4\uFF08\u5305\u62EC\u5E38\u89C1\u7684\u6B27\u51E0\u91CC\u5F97\u7A7A\u95F4\uFF09\u4E2D\u7684\u4E00\u7EC4\u7EBF\u6027\u65E0\u5173\u5411\u91CFv1\uFF0C...\uFF0Cvk\u51FA\u53D1\uFF0C\u5F97\u5230\u540C\u4E00\u4E2A\u5B50\u7A7A\u95F4\u4E0A\u4E24\u4E24\u6B63\u4EA4\u7684\u5411\u91CF\u7EC4u1\uFF0C...\uFF0Cuk\u3002 \u5982\u679C\u8FD8\u8981\u6C42\u6B63\u4EA4\u5316\u540E\u7684\u5411\u91CF\u90FD\u662F\u5355\u4F4D\u5411\u91CF\uFF0C\u90A3\u4E48\u79F0\u4E3A\u6807\u51C6\u6B63\u4EA4\u5316\u3002 \u4E00\u822C\u5728\u6570\u5B66\u5206\u6790\u4E2D\u91C7\u7528\u683C\u62C9\u59C6-\u65BD\u5BC6\u7279\u6B63\u4EA4\u5316\u4F5C\u6B63\u4EA4\u5316\u7684\u8BA1\u7B97\u3002\u5728\u7F16\u7A0B\u8BA1\u7B97\u65F6\uFF0C\u683C\u62C9\u59C6-\u65BD\u5BC6\u7279\u6B63\u4EA4\u5316\u7684\u6570\u503C\u7A33\u5B9A\u6027\u4E0D\u9AD8\uFF0C\u6240\u4EE5\u5E38\u7528\u66F4\u7A33\u5B9A\u7684\u8C6A\u65AF\u970D\u5C14\u5FB7\u53D8\u6362\u4EE3\u66FF\u3002\u53E6\u5916\uFF0C\u76F8\u5BF9\u4E8E\u8C6A\u65AF\u970D\u5C14\u5FB7\u53D8\u6362\u5728\u6700\u540E\u76F4\u63A5\u751F\u6210\u6240\u6709\u7684\u5411\u91CF\uFF0C\u683C\u62C9\u59C6-\u65BD\u5BC6\u7279\u65B9\u6CD5\u5728\u7B2Ci\u6B65\u4EA7\u751F\u7B2Ci\u4E2A\u5411\u91CF\uFF0C\u56E0\u6B64\u540E\u8005\u53EF\u7528\u8FED\u4EE3\u6CD5\u7F16\u5199\u3002\u5BF9\u4E8E\u542B\u6709\u96F6\u5143\u7D20\u8F83\u591A\u7684\u5411\u91CF\u7EC4\uFF08\u4F8B\u5982\u7A00\u758F\u77E9\u9635\u7684QR\u5206\u89E3\uFF09\uFF0C\u8FD8\u4F1A\u91C7\u7528\u5409\u6587\u65AF\u65CB\u8F6C\u3002"@zh . . . . "Mit Orthogonalisierungsverfahren bezeichnet man in der Mathematik Algorithmen, die aus einem System linear unabh\u00E4ngiger Vektoren ein Orthogonalsystem erzeugen, das den gleichen Untervektorraum aufspannt. mit einer orthogonalen Matrix und einer Dreiecksmatrix zu erzeugen. Die Spaltenvektoren der Matrix sind dann die orthogonalisierten Spaltenvektoren der Matrix . Haupts\u00E4chlich erh\u00E4lt man aber eine stabile Methode zum L\u00F6sen linearer Gleichungssysteme."@de . "\u76F4\u4EA4\u5316\uFF08\u3061\u3087\u3063\u3053\u3046\u304B\uFF09\u3068\u306F\u3001\u7DDA\u578B\u7A7A\u9593\u4E0A\u306B\u3042\u308B\u30D9\u30AF\u30C8\u30EB\u306E\u7D44\u304B\u3089\u3001\u4E92\u3044\u306B\u76F4\u4EA4\u3059\u308B\u30D9\u30AF\u30C8\u30EB\u306E\u7D44\u3092\u751F\u6210\u3059\u308B\u3053\u3068\u3067\u3042\u308B\u3002"@ja . . . "\u041E\u0440\u0442\u043E\u0433\u043E\u043D\u0430\u043B\u0438\u0437\u0430\u0446\u0438\u044F \u2015 \u043F\u0440\u043E\u0446\u0435\u0441\u0441 \u043F\u043E\u0441\u0442\u0440\u043E\u0435\u043D\u0438\u044F \u043F\u043E \u0437\u0430\u0434\u0430\u043D\u043D\u043E\u043C\u0443 \u0431\u0430\u0437\u0438\u0441\u0443 \u043B\u0438\u043D\u0435\u0439\u043D\u043E\u0433\u043E \u043F\u0440\u043E\u0441\u0442\u0440\u0430\u043D\u0441\u0442\u0432\u0430 \u043D\u0435\u043A\u043E\u0442\u043E\u0440\u043E\u0433\u043E \u043E\u0440\u0442\u043E\u0433\u043E\u043D\u0430\u043B\u044C\u043D\u043E\u0433\u043E \u0431\u0430\u0437\u0438\u0441\u0430, \u043A\u043E\u0442\u043E\u0440\u044B\u0439 \u0438\u043C\u0435\u0435\u0442 \u0442\u0443 \u0436\u0435 \u0441\u0430\u043C\u0443\u044E \u043B\u0438\u043D\u0435\u0439\u043D\u0443\u044E \u043E\u0431\u043E\u043B\u043E\u0447\u043A\u0443. \u0412\u0432\u0438\u0434\u0443 \u0443\u0434\u043E\u0431\u0441\u0442\u0432\u0430 \u0438 \u0432\u0430\u0436\u043D\u043E\u0441\u0442\u0438 \u043E\u0440\u0442\u043E\u0433\u043E\u043D\u0430\u043B\u044C\u043D\u044B\u0445 \u0431\u0430\u0437\u0438\u0441\u043E\u0432 \u0432 \u0440\u0430\u0437\u043B\u0438\u0447\u043D\u044B\u0445 \u0437\u0430\u0434\u0430\u0447\u0430\u0445, \u0432\u0430\u0436\u043D\u044B \u0438 \u043F\u0440\u043E\u0446\u0435\u0441\u0441\u044B \u043E\u0440\u0442\u043E\u0433\u043E\u043D\u0430\u043B\u0438\u0437\u0430\u0446\u0438\u0438."@ru .