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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 Haynsworth inertia additivity formula ： ウィキペディア英語版 Haynsworth inertia additivity formula In mathematics, the Haynsworth inertia additivity formula, discovered by Emilie Virginia Haynsworth〔Haynsworth, E. V., "Determination of the inertia of a partitioned Hermitian matrix", ''Linear Algebra and its Applications'', volume 1 (1968), pages 73–81〕 (1916–1985), concerns the number of positive, negative, and zero eigenvalues of a Hermitian matrix and of block matrices into which it is partitioned. The ''inertia'' of a Hermitian matrix ''H'' is defined as the ordered triple : $\backslash mathrm(H)\; =\; \backslash left(\; \backslash pi(H),\; \backslash nu(H),\; \backslash delta(H)\; \backslash right)\; \backslash ,$ whose components are respectively the numbers of positive, negative, and zero eigenvalues of ''H''. Haynsworth considered a partitioned Hermitian matrix : where ''H''11 is nonsingular and ''H''12 * is the conjugate transpose of ''H''12. The formula states:() : where ''H''/''H''11 is the Schur complement of ''H''11 in ''H'': : $H/H\_\; =\; H\_\; -\; H\_^\backslash ast\; H\_^H\_.\; \backslash ,$ == Generalization == If ''H''11 is singular, we can still define the generalized Schur complement, using the Moore–Penrose inverse instead of . The formula does not hold if ''H''11 is singular. However, a generalization has been proven in 1974 by Carlson, Haynsworth and Markham,〔D. Carlson, E. V. Haynsworth, and T. Markham, "A generalization of the Schur complement by means of the Moore–Penrose inverse", ''SIAM J. Appl. Math.'', volume 16(1) (1974), pages 169–175〕 to the effect that $\backslash pi(H)\; \backslash ge\; \backslash pi(H\_)\; +\; \backslash pi(H/H\_)$ and $\backslash nu(H)\; \backslash ge\; \backslash nu(H\_)\; +\; \backslash nu(H/H\_)$. Carlson, Haynsworth and Markham also gave sufficient and necessary conditions for equality to hold. 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「Haynsworth inertia additivity formula」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.