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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 Jacobi method for complex Hermitian matrices ： ウィキペディア英語版 Jacobi method for complex Hermitian matrices In mathematics, the Jacobi method for complex Hermitian matrices is a generalization of the Jacobi iteration method. The Jacobi iteration method is also explained in "Introduction to Linear Algebra" by . == Derivation == The complex unitary rotation matrices ''R''''pq'' can be used for Jacobi iteration of complex Hermitian matrices in order to find a numerical estimation of their eigenvectors and eigenvalues simultaneously. Similar to the Givens rotation matrices, ''R''''pq'' are defined as: :$$ \begin (R_)_ & = \delta_ & \qquad m,n \ne p,q, \\() (R_)_ & = \frac, \\() (R_)_ & = \frac, \\() (R_)_ & = \frac, \\() (R_)_ & = \frac \end Each rotation matrix, ''R''''pq'', will modify only the ''p''th and ''q''th rows or columns of a matrix ''M'' if it is applied from left or right, respectively: :$$ \begin (R_ M)_ & = \begin M_ & m \ne p,q \\() \frac e^ - M_ e^) & m = p \\() \frac e^ + M_ e^) & m = q \end \\() (MR_^\dagger)_ & = \begin M_ & n \ne p,q \\ \frac e^ - M_ e^) & n = p \\() \frac e^ + M_ e^) & n = q \end \end A Hermitian matrix, ''H'' is defined by the conjugate transpose symmetry property: :$H^\backslash dagger\; =\; H\; \backslash \; \backslash Leftrightarrow\backslash \; H\_\; =\; H^\_$ By definition, the complex conjugate of a complex unitary rotation matrix, ''R'' is its inverse and also a complex unitary rotation matrix: :$$ \begin R^\dagger_ & = R^_ \\() \Rightarrow\ R^_ & = R^_ = R^ = R_. \end Hence, the complex equivalent Givens transformation $T$ of a Hermitian matrix ''H'' is also a Hermitian matrix similar to ''H'': :$$ \begin T & \equiv R_ H R^\dagger_, & & \\() T^\dagger & = (R_ H R^\dagger_)^\dagger = R^_ H^\dagger R^\dagger_ = R_ H R^\dagger_ = T \end The elements of ''T'' can be calculated by the relations above. The important elements for the Jacobi iteration are the following four: :$$ \begin T_ & = & & \frac} & - \ \ \ \mathrm\\}, \\() T_ & = & & \frac} & + \ i \ \mathrm\\}, \\() T_ & = & & \frac} & - \ i \ \mathrm\\}, \\() T_ & = & & \frac} & + \ \ \ \mathrm\\}. \end Each Jacobi iteration with ''R''''J''''pq'' generates a transformed matrix, ''T''''J'', with ''T''''J''''p'',''q'' = 0. The rotation matrix ''R''''J''''p'',''q'' is defined as a product of two complex unitary rotation matrices. :$$ \begin R^J_ & \equiv R_(\theta_2)\, R_(\theta_1),\text \\() \theta_1 & \equiv \frac \text \theta_2 \equiv \frac, \end where the phase terms, $\backslash phi\_1$ and $\backslash phi\_2$ are given by: :$$ \begin \tan \phi_1 & = \frac\}}\}}, \\() \tan \phi_2 & = \frac}. \end Finally, it is important to note that the product of two complex rotation matrices for given angles ''θ''1 and ''θ''2 cannot be transformed into a single complex unitary rotation matrix ''R''''pq''(''θ''). The product of two complex rotation matrices are given by: :$$ \begin \left(R_(\theta_2)\, R_(\theta_1) \right )_ = \begin \ \ \ \ \delta_ & m,n \ne p,q, \\() -i e^\, \sin & m = p \text n = p, \\() - e^\, \cos & m = p \text n = q, \\() \ \ \ \ e^\, \cos & m = q \text n = p, \\() +i e^\, \sin & m = q \text n = q. \end \end 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「Jacobi method for complex Hermitian 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.