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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 2 × 2 real matrices ： ウィキペディア英語版 2 × 2 real matrices In mathematics, the set of real matrices is denoted by M(2, R). Two matrices ''p'' and ''q'' in M(2, R) have a sum ''p'' + ''q'' given by matrix addition. The product matrix is formed from the dot product of the rows and columns of its factors through matrix multiplication. For : let : Then ''q q'' * = ''q'' * ''q'' = (''ad'' − ''bc'') , where is the identity matrix. The real number ''ad'' − ''bc'' is called the determinant of ''q''. When ''ad'' − ''bc'' ≠ 0, ''q'' is an invertible matrix, and then :$q^\; =\; q^$ *\,/\,(ad - bc). The collection of all such invertible matrices constitutes the general linear group GL(2, R). In terms of abstract algebra, M(2, R) with the associated addition and multiplication operations forms a ring, and GL(2, R) is its group of units. M(2, R) is also a four-dimensional vector space, so it is considered an associative algebra. It is ring-isomorphic to the coquaternions, but has a different profile. The real matrices are in one-one correspondence with the linear mappings of the two-dimensional Cartesian coordinate system into itself by the rule : \beginax + by \\ cx + dy\end. ==Profile== Within M(2, R), the multiples by real numbers of the identity matrix may be considered a real line. This real line is the place where all commutative subrings come together: Let ''P''''m'' =  where ''m''2 ∈ . Then ''P''''m'' is a commutative subring and M(2, R) = ∪''P''''m''   where the union is over all ''m'' such that ''m''2 ∈ . To identify such ''m'', first square the generic matrix: : When ''a'' + ''d'' = 0 this square is a diagonal matrix. Thus one assumes ''d'' = −''a'' when looking for ''m'' to form commutative subrings. When ''mm'' = −, then ''bc'' = −1 − ''aa'', an equation describing a hyperbolic paraboloid in the space of parameters (''a'', ''b'', ''c''). Such an ''m'' serves as an imaginary unit. In this case P''m'' is isomorphic to the field of (ordinary) complex numbers. When ''mm'' = +, ''m'' is an involutory matrix. Then ''bc'' = +1 − ''aa'', also giving a hyperbolic paraboloid. If a matrix is an idempotent matrix, it must lie in such a P''m'' and in this case P''m'' is isomorphic to the ring of split-complex numbers. The case of a nilpotent matrix, ''mm'' = 0, arises when only one of ''b'' or ''c'' is non-zero, and the commutative subring P''m'' is then a copy of the dual number plane. When M(2, R) is reconfigured with a change of basis, this profile changes to the profile of split-quaternions where the sets of square roots of and − take a symmetrical shape as hyperboloids. 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「2 × 2 real matrices」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.