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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 Unitary group ： ウィキペディア英語版 Unitary group In mathematics, the unitary group of degree ''n'', denoted U(''n''), is the group of unitary matrices, with the group operation that of matrix multiplication. The unitary group is a subgroup of the general linear group . Hyperorthogonal group is an archaic name for the unitary group, especially over finite fields. In the simple case , the group U(1) corresponds to the circle group, consisting of all complex numbers with absolute value 1 under multiplication. All the unitary groups contain copies of this group. The unitary group U(''n'') is a real Lie group of dimension ''n''2. The Lie algebra of U(''n'') consists of skew-Hermitian matrices, with the Lie bracket given by the commutator. The general unitary group (also called the group of unitary similitudes) consists of all matrices ''A'' such that ''A''∗''A'' is a nonzero multiple of the identity matrix, and is just the product of the unitary group with the group of all positive multiples of the identity matrix. ==Properties== Since the determinant of a unitary matrix is a complex number with norm $1$, the determinant gives a group homomorphism :$\backslash det\; \backslash colon\; \backslash operatorname(n)\; \backslash to\; \backslash operatorname(1).$ The kernel of this homomorphism is the set of unitary matrices with determinant $1$. This subgroup is called the special unitary group, denoted $\backslash operatorname(n)$. We then have a short exact sequence of Lie groups: :$1\; \backslash to\; \backslash operatorname(n)\; \backslash to\; \backslash operatorname(n)\; \backslash to\; \backslash operatorname(1)\; \backslash to\; 1.$ This short exact sequence splits so that $\backslash operatorname(n)$ may be written as a semidirect product of $\backslash operatorname(n)$ by $\backslash operatorname(1)$. Here the $\backslash operatorname(1)$ subgroup of $\backslash operatorname(n)$ can be taken to consist of matrices, which are diagonal, have $e^$ in the upper left corner and $1$ on the rest of the diagonal. The unitary group $\backslash operatorname(n)$ is nonabelian for $n\; >\; 1$. The center of $\backslash operatorname(n)$ is the set of scalar matrices $\backslash lambda\; I$ with $\backslash lambda\; \backslash in\; \backslash operatorname(1)$. This follows from Schur's lemma. The center is then isomorphic to $\backslash operatorname(1)$. Since the center of $\backslash operatorname(n)$ is a $1$-dimensional abelian normal subgroup of $\backslash operatorname(n)$, the unitary group is not semisimple. 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「Unitary group」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.