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In linear algebra, the Frobenius companion matrix of the monic polynomial : is the square matrix defined as : With this convention, and on the basis , one has : (for ), and generates as a -module: cycles basis vectors. Some authors use the transpose of this matrix, which (dually) cycles coordinates, and is more convenient for some purposes, like linear recurrence relations. ==Characterization== The characteristic polynomial as well as the minimal polynomial of are equal to .〔 〕 In this sense, the matrix is the "companion" of the polynomial . If is an ''n''-by-''n'' matrix with entries from some field , then the following statements are equivalent: * is similar to the companion matrix over of its characteristic polynomial * the characteristic polynomial of coincides with the minimal polynomial of , equivalently the minimal polynomial has degree * there exists a cyclic vector in for , meaning that is a basis of ''V''. Equivalently, such that ''V'' is cyclic as a -module (and ); one says that is ''regular''. Not every square matrix is similar to a companion matrix. But every matrix is similar to a matrix made up of blocks of companion matrices. Furthermore, these companion matrices can be chosen so that their polynomials divide each other; then they are uniquely determined by . This is the rational canonical form of . 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「Companion matrix」の詳細全文を読む スポンサード リンク
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