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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 Complex Hadamard matrix ： ウィキペディア英語版 Complex Hadamard matrix A complex Hadamard matrix is any complex $N\; \backslash times\; N$ matrix $H$ satisfying two conditions: *unimodularity (the modulus of each entry is unity): $|H\_|=1\; j,k=1,2,\backslash dots,N$ *orthogonality: $HH^\; =\; N\; \backslash ;$, where $$ denotes the Hermitian transpose of ''H'' and $$ is the identity matrix. The concept is a generalization of the Hadamard matrix. Note that any complex Hadamard matrix $H$ can be made into a unitary matrix by multiplying it by $\backslash frac$. Complex Hadamard matrices arise in the study of operator algebras and the theory of quantum computation. Real Hadamard matrices and Butson-type Hadamard matrices form particular cases of complex Hadamard matrices. Complex Hadamard matrices exist for any natural ''N'' (compare the real case, in which existence is not known for every ''N''). For instance the Fourier matrices :$()\_:=\; \backslash exp(i(j\; -\; 1)(k\; -\; 1)\; /\; N)$ j,k=1,2,\dots,N belong to this class. ==Equivalency== Two complex Hadamard matrices are called equivalent, written $H\_1\; \backslash simeq\; H\_2$, if there exist diagonal unitary matrices $D\_1,\; D\_2$ and permutation matrices $P\_1,\; P\_2$ such that :$H\_1\; =\; D\_1\; P\_1\; H\_2\; P\_2\; D\_2.$ Any complex Hadamard matrix is equivalent to a dephased Hadamard matrix, in which all elements in the first row and first column are equal to unity. For $N=2,3$ and $5$ all complex Hadamard matrices are equivalent to the Fourier matrix $F\_$. For $N=4$ there exists a continuous, one-parameter family of inequivalent complex Hadamard matrices, :$F\_^(a):=$ \begin 1 & 1 & 1 & 1 \\ 1 & ie^ & -1 & -ie^ \\ 1 & -1 & 1 &-1 \\ 1 & -ie^& -1 & i e^ \end a\in [0,\pi) . For $N=6$ the following families of complex Hadamard matrices are known: * a single two-parameter family which includes $F\_6$, * a single one-parameter family $D\_6(t)$, * a one-parameter orbit $B\_6(\backslash theta)$, including the circulant Hadamard matrix $C\_6$, * a two-parameter orbit including the previous two examples $X\_6(\backslash alpha)$, * a one-parameter orbit $M\_6(x)$ of symmetric matrices, * a two-parameter orbit including the previous example $K\_6(x,y)$, * a three-parameter orbit including all the previous examples $K\_6(x,y,z)$, * a further construction with four degrees of freedom, $G\_6$, yielding other examples than $K\_6(x,y,z)$, * a single point - one of the Butson-type Hadamard matrices, $S\_6\; \backslash in\; H(3,6)$. It is not known, however, if this list is complete, but it is conjectured that $K\_6(x,y,z),G\_6,S\_6$ is an exhaustive (but not necessarily irredundant) list of all complex Hadamard matrices of order 6. 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「Complex Hadamard matrix」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.