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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 sedenion ： ウィキペディア英語版 sedenion In abstract algebra, the sedenions form a 16-dimensional noncommutative and nonassociative algebra over the reals obtained by applying the Cayley–Dickson construction to the octonions. The set of sedenions is denoted by $\backslash mathbb$. The term "sedenion" is also used for other 16-dimensional algebraic structures, such as a tensor product of 2 copies of the biquaternions, or the algebra of 4 by 4 matrices over the reals, or that studied by . ==Arithmetic== Like octonions, multiplication of sedenions is neither commutative nor associative. But in contrast to the octonions, the sedenions do not even have the property of being alternative. They do, however, have the property of power associativity, which can be stated as for any element x of $\backslash mathbb$, the power $x^n$ is well-defined. They are also flexible. Every sedenion is a linear combination of the unit sedenions e0, e1, e2, e3, ...,e15, which form a basis of the vector space of sedenions. Every sedenion can be represented in the form :$x\; =\; x\_0e\_0\; +\; x\_1e\_1\; +\; x\_2e\_2\; +\; \backslash ldots\; +\; x\_e\_\; +\; x\_e\_,\backslash ,$. Addition and subtraction are defined by the addition and subtraction of corresponding coefficients and multiplication is distributive over addition. Like other algebras based on the Cayley–Dickson construction, the sedenions contain the algebra it was constructed from. So they contain the octonions (e0 to e7 in the table below), and therefore also the quaternions (e0 to e3), complex numbers (e0 and e1) and reals (e0). The sedenions have a multiplicative identity element e0 and multiplicative inverses but they are not a division algebra because they have zero divisors. This means that two non-zero sedenions can be multiplied to obtain zero: an example is (e3 + e10)×(e6 − e15). All hypercomplex number systems based on the Cayley–Dickson construction after sedenions contain zero divisors. The multiplication table of these unit sedenions follows: From the above table, we can see that: :$e\_0e\_i\; =\; e\_ie\_0\; =\; e\_i,$ :$e\_ie\_i\; =\; -e\_0\; \backslash ,\backslash ,\; \backslash text\backslash ,\backslash ,\; i\; \backslash neq\; 0,$ :$e\_ie\_j\; =\; -e\_je\_i\; \backslash ,\backslash ,\; \backslash text\backslash ,\backslash ,\; i\; \backslash neq\; j\; \backslash ,\backslash ,\backslash text\backslash ,\backslash ,\; i,j\; \backslash neq\; 0.$ :$e\_i(e\_je\_k)\; =\; -(e\_ie\_j)e\_k\; \backslash ,\backslash ,\backslash text\backslash ,\backslash ,\; i\; \backslash neq\; j,\backslash ,\backslash ,\; i,j\; \backslash neq\; 0\backslash ,\backslash ,\backslash text\backslash ,\backslash ,\; e\_ie\_j\backslash neq\; \backslash plusmn\; e\_k.$ 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「sedenion」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.