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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 Bose–Mesner algebra ： ウィキペディア英語版 Bose–Mesner algebra In mathematics, a Bose–Mesner algebra is a special set of matrices which arise from a combinatorial structure known as an association scheme, together with the usual set of rules for combining (forming the products of) those matrices, such that they form an associative algebra, or, more precisely, a unitary commutative algebra. Among these rules are: : *the result of a product is also within the set of matrices, : *there is an identity matrix in the set, and : *taking products is commutative. Bose–Mesner algebras have applications in physics to spin models, and in statistics to the design of experiments. They are named for R. C. Bose and Dale Marsh Mesner.〔Bose & Mesner (1959)〕 ==Definition== Let ''X'' be a set of ''v'' elements. Consider a partition of the 2-element subsets of ''X'' into ''n'' non-empty subsets, ''R''1, ..., ''R''''n'' such that: * given an $x\; \backslash in\; X$, the number of $y\; \backslash in\; X$ such that $\backslash \; \backslash in\; R\_i$ depends only on i (and not on ''x''). This number will be denoted by vi, and * given $x,y\; \backslash in\; X$ with $\backslash \; \backslash in\; R\_k$, the number of $z\; \backslash in\; X$ such that $\backslash \; \backslash in\; R\_i$ and $\backslash \; \backslash in\; R\_j$ depends only on ''i'',''j'' and ''k'' (and not on ''x'' and ''y''). This number will be denoted by $p^k\_$. This structure is enhanced by adding all pairs of repeated elements of ''X'' and collecting them in a subset ''R''0. This enhancement permits the parameters ''i'', ''j'', and ''k'' to take on the value of zero, and lets some of ''x'',''y'' or ''z'' be equal. A set with such an enhanced partition is called an Association scheme. One may view an association scheme as a partition of the edges of a complete graph (with vertex set ''X'') into n classes, often thought of as color classes. In this representation, there is a loop at each vertex and all the loops receive the same 0th color. The association scheme can also be represented algebraically. Consider the matrices ''D''''i'' defined by: : $(D\_i)\_\; =\; \backslash begin$ 1,& \text \left(x,y\right)\in R_,\\ 0,& \text \end \qquad (1) Let $\backslash mathcal$ be the vector space consisting of all matrices $\backslash sideset\}\backslash sum\; a\_D\_$, with $a\_$ complex. The definition of an association scheme is equivalent to saying that the $D\_$ are ''v'' × ''v'' (0,1)-matrices which satisfy # $D\_i$ is symmetric, # $\backslash sum\_^n\; D\_=J$ (the all-ones matrix), # $D\_0=I,$ # $D\_i\; D\_j\; =\; \backslash sum\_^n\; p^k\_\; D\_k\; =\; D\_j\; D\_i,\backslash qquad\; i,j=0,\backslash ldots,n.$ The (''x'',''y'')-th entry of the left side of 4. is the number of two colored paths of length two joining ''x'' and ''y'' (using "colors" ''i'' and ''j'') in the graph. Note that the rows and columns of $D\_i$ contain $v\_i$ 1s: : $D\_i\; J=J\; D\_i\; =\; v\_i\; J.\; \backslash qquad\; (2)$ From 1., these matrices are symmetric. From 2., $D\_,\backslash ldots,D\_$ are linearly independent, and the dimension of $\backslash mathcal$ is $n+1$. From 4., $\backslash mathcal$ is closed under multiplication, and multiplication is always associative. This associative commutative algebra $\backslash mathcal$ is called the Bose–Mesner algebra of the association scheme. Since the matrices in $\backslash mathcal$ are symmetric and commute with each other, they can be simultaneously diagonalized. This means that there is a matrix $S$ such that to each $A\backslash in\backslash mathcal$ there is a diagonal matrix $\backslash Lambda\_$ with $S^A\; S=\backslash Lambda\_$. This means that $\backslash mathcal$ is semi-simple and has a unique basis of primitive idempotents $J\_,\backslash ldots,J\_$. These are complex n × n matrices satisfying : $$ J_i^2 =J_i, i=0,\ldots,n, \qquad (3) : $$ J_i J_k=0, i\neq k, \qquad (4) : $$ \sum_^n J_i = I. \qquad (5) The Bose–Mesner algebra has two distinguished bases: the basis consisting of the adjacency matrices $D\_i$, and the basis consisting of the irreducible idempotent matrices $E\_k$. By definition, there exist well-defined complex numbers such that : $$ D_=\sum_^n p_i (k) E_k, \qquad (6) and : $$ |X|E_=\sum_^n q_k\left(i\right)D_i. \qquad (7) The p-numbers $p\_i\; (k)$, and the q-numbers $q\_k(i)$, play a prominent role in the theory. They satisfy well-defined orthogonality relations. The p-numbers are the eigenvalues of the adjacency matrix $D\_i$. 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「Bose–Mesner algebra」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.