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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 Matrix pencil ： ウィキペディア英語版 Matrix pencil In linear algebra, if $A\_0,\; A\_1,\backslash dots,A\_l$ are $n\backslash times\; n$ complex matrices for some nonnegative integer $l$, and $A\_l\; \backslash ne\; 0$ (the zero matrix), then the matrix pencil of degree $l$ is the matrix-valued function defined on the complex numbers :$L(\backslash lambda)\; =\; \backslash sum\_^l\; \backslash lambda^i\; A\_i.$ A particular case is a linear matrix pencil :$A-\backslash lambda\; B\; \backslash ,$ with :$\backslash lambda\; \backslash in\; \backslash mathbb\; C\backslash text\backslash mathbb\; R\backslash text$ where $A$ and $B$ are complex (or real) $n\; \backslash times\; n$ matrices. We denote it briefly with the notation $(A,B)$. A pencil is called ''regular'' if there is at least one value of $\backslash lambda$ such that $\backslash det(A-\backslash lambda\; B)\backslash neq\; 0$. We call ''eigenvalues'' of a matrix pencil $(A,B)$ all complex numbers $\backslash lambda$ for which $\backslash det(A-\backslash lambda\; B)=0$ (see eigenvalue for comparison). The set of the eigenvalues is called the ''spectrum'' of the pencil and is written $\backslash sigma(A,B)$. Moreover, the pencil is said to have one or more eigenvalues at infinity if $B$ has one or more 0 eigenvalues. ==Applications== Matrix pencils play an important role in numerical linear algebra. The problem of finding the eigenvalues of a pencil is called the generalized eigenvalue problem. The most popular algorithm for this task is the QZ algorithm, which is an implicit version of the QR algorithm to solve the associated eigenvalue problem $B^Ax=\backslash lambda\; x$ without forming explicitly the matrix $B^A$ (which could be impossible or ill-conditioned if $B$ is singular or near-singular) 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「Matrix pencil」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.