翻訳と辞書 Words near each other ・ Arrowhead (train) ・ Arrowhead Christian Academy ・ Arrowhead Conference ・ Arrowhead Council ・ Arrowhead device ・ Arrowhead dogfish ・ Arrowhead Farms, San Bernardino, California ・ Arrowhead Game Studios ・ Arrowhead High School ・ Arrowhead Lake (Idaho) ・ Arrowhead Lake (Ontario) ・ Arrowhead League ・ Arrowhead Library System ・ Arrowhead Mall ・ Arrowhead Marsh ・ Arrowhead matrix ・ Arrowhead Mills ・ Arrowhead Monument ・ Arrowhead Mountain ・ Arrowhead Nunatak ・ Arrowhead Pawn Shop ・ Arrowhead piculet ・ Arrowhead Pool ・ Arrowhead Provincial Park ・ Arrowhead Range ・ Arrowhead Recreation Area ・ Arrowhead Refinery Company ・ Arrowhead Region ・ Arrowhead Regional Medical Center ・ Arrowhead Springs, San Bernardino, California Dictionary Lists mini英和辞書 mini和英辞書 Webster 1913 Latin-English FOLDOC Wikipedia English ウィキペディア

翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Arrowhead matrix ： ウィキペディア英語版 Arrowhead matrix In the mathematical field of linear algebra, an arrowhead matrix is a square matrix containing zeros in all entries except for the first row, first column, and main diagonal. In other words, the matrix has the form :$$ A=\begin \,\! *& *& *& *& * \\ \,\! *& *&0&0&0 \\ \,\! *&0& *&0&0 \\ \,\! *&0&0& *&0 \\ \,\! *&0&0&0& * \end. Any symmetric permutation of the arrowhead matrix, $P^T\; A\; P$, where ''P'' is a permutation matrix, is a (permuted) arrowhead matrix. Real symmetric arrowhead matrices are used in some algorithms for finding of eigenvalues and eigenvectors. ==Real symmetric arrowhead matrices== Let ''A'' be a real symmetric (permuted) arrowhead matrix of the form :$$ A=\left(& z \\ z^ & \alpha \end \right ), where $D=\backslash mathop,d\_,\backslash ldots\; ,d\_)$ is diagonal matrix of order ''n-1'', $z=\backslash begin$ \zeta _ & \zeta _ & \cdots & \zeta _ \end^T is a vector and $\backslash alpha$ is a scalar. Let :$$ A=V\Lambda V^ be the eigenvalue decomposition of ''A'', where is an orthonormal matrix whose columns are the corresponding eigenvectors. The following holds: * If $\backslash zeta\_i=0$ for some ''i'', then the pair $(d\_i,e\_i)$, where $e\_i$ is the ''i''-th standard basis vector, is an eigenpair of ''A''. Thus, all such rows and columns can be deleted, leaving the matrix with all $\backslash zeta\_i\backslash neq\; 0$. * The Cauchy interlacing theorem implies that the sorted eigenvalues of ''A'' interlace the sorted elements $d\_i$: if $d\_1\; \backslash geq\; d\_2\backslash geq\; \backslash cdots\backslash geq\; d\_$ (this can be attained by symmetric permutation of rows and columns without loss of generality), and if $\backslash lambda\_i$s are sorted accordingly, then $$ \lambda_1\geq d_1\geq \lambda_2\geq d_2\geq \cdots \geq \lambda_ \geq d_ \geq \lambda_n . * If $d\_i=d\_j$, for some $i\backslash neq\; j$, the above inequality implies that $d\_$ is an eigenvalue of ''A''. The size of the problem can be reduced by annihilating $\backslash zeta\_j$ with a Givens rotation in the $(i,j)$-plane and proceeding as above. 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「Arrowhead 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.