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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 , the determinant gives a group homomorphism : The kernel of this homomorphism is the set of unitary matrices with determinant . This subgroup is called the special unitary group, denoted . We then have a short exact sequence of Lie groups: : This short exact sequence splits so that may be written as a semidirect product of by . Here the subgroup of can be taken to consist of matrices, which are diagonal, have in the upper left corner and on the rest of the diagonal. The unitary group is nonabelian for . The center of is the set of scalar matrices with . This follows from Schur's lemma. The center is then isomorphic to . Since the center of is a -dimensional abelian normal subgroup of , 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
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