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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 majorization ： ウィキペディア英語版 majorization In mathematics, majorization is a preorder on vectors of real numbers. For a vector $\backslash mathbf\backslash in\backslash mathbb^d$, we denote by $\backslash mathbf^\backslash in\backslash mathbb^d$ the vector with the same components, but sorted in descending order. Given $\backslash mathbf,\backslash mathbf\; \backslash in\; \backslash mathbb^d$, we say that $\backslash mathbf$ weakly majorizes (or dominates) $\backslash mathbf$ from below written as $\backslash mathbf\; \backslash succ\_w\; \backslash mathbf$ iff : $\backslash sum\_^k\; a\_i^\; \backslash geq\; \backslash sum\_^k\; b\_i^\; \backslash quad\; \backslash text\; k=1,\backslash dots,d,$ where $a^\_i$ and $b^\_i$ are the elements of $\backslash mathbf$ and $\backslash mathbf$, respectively, sorted in decreasing order. Equivalently, we say that $\backslash mathbf$ is weakly majorized (or dominated) by $\backslash mathbf$ from below, denoted as $\backslash mathbf\; \backslash prec\_w\; \backslash mathbf$. Similarly, we say that $\backslash mathbf$ weakly majorizes $\backslash mathbf$ from above written as $\backslash mathbf\; \backslash succ^w\; \backslash mathbf$ iff : $\backslash sum\_^d\; a\_i^\; \backslash leq\; \backslash sum\_^d\; b\_i^\; \backslash quad\; \backslash text\; k=1,\backslash dots,d,$ Equivalently, we say that $\backslash mathbf$ is weakly majorized by $\backslash mathbf$ from above, denoted as $\backslash mathbf\; \backslash prec^w\; \backslash mathbf$. If $\backslash mathbf\; \backslash succ\_w\; \backslash mathbf$ and in addition $\backslash sum\_^d\; a\_i\; =\; \backslash sum\_^d\; b\_i$ we say that $\backslash mathbf$ majorizes (or dominates) $\backslash mathbf$ written as $\backslash mathbf\; \backslash succ\; \backslash mathbf$. Equivalently, we say that $\backslash mathbf$ is majorized (or dominated) by $\backslash mathbf$, denoted as $\backslash mathbf\; \backslash prec\; \backslash mathbf$. It is easy to see that $\backslash mathbf\; \backslash succ\; \backslash mathbf$ if and only if $\backslash mathbf\; \backslash succ\_w\; \backslash mathbf$ and $\backslash mathbf\; \backslash succ^w\; \backslash mathbf$. Note that the majorization order do not depend on the order of the components of the vectors $\backslash mathbf$ or $\backslash mathbf$. Majorization is not a partial order, since $\backslash mathbf\; \backslash succ\; \backslash mathbf$ and $\backslash mathbf\; \backslash succ\; \backslash mathbf$ do not imply $\backslash mathbf\; =\; \backslash mathbf$, it only implies that the components of each vector are equal, but not necessarily in the same order. Regrettably, to confuse the matter, some literature sources use the reverse notation, e.g., $\backslash succ$ is replaced with $\backslash prec$, most notably, in Horn and Johnson, Matrix analysis (Cambridge Univ. Press, 1985), Definition 4.3.24, while the same authors switch to the traditional notation, introduced here, later in their ''Topics in Matrix Analysis'' (1994). A function $f:\backslash mathbb^d\; \backslash to\; \backslash mathbb$ is said to be Schur convex when $\backslash mathbf\; \backslash succ\; \backslash mathbf$ implies $f(\backslash mathbf)\; \backslash geq\; f(\backslash mathbf)$. Similarly, $f(\backslash mathbf)$ is Schur concave when $\backslash mathbf\; \backslash succ\; \backslash mathbf$ implies $f(\backslash mathbf)\; \backslash leq\; f(\backslash mathbf).$ The majorization partial order on finite sets, described here, can be generalized to the Lorenz ordering, a partial order on distribution functions. ==Examples== The order of the entries does not affect the majorization, e.g., the statement $(1,2)\backslash prec\; (0,3)$ is simply equivalent to $(2,1)\backslash prec\; (3,0)$. (Strong) majorization: $(1,2,3)\backslash prec\; (0,3,3)\backslash prec\; (0,0,6)$. For vectors with ''n'' components : $$ \left(\frac, \ldots, \frac\right)\prec \left(\frac, \ldots, \frac,0\right) \prec \cdots \prec \left(\frac,\frac, 0, \ldots, 0\right) \prec \left(1, 0, \ldots, 0\right). (Weak) majorization: $(1,2,3)\backslash prec\_w\; (1,3,3)\backslash prec\_w\; (1,3,4)$. For vectors with ''n'' components: : $$ \left(\frac, \ldots, \frac\right)\prec_w \left(\frac, \ldots, \frac,1\right). 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「majorization」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.