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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 polynomial ： ウィキペディア英語版 Matrix polynomial In mathematics, a matrix polynomial is a polynomial with matrices as variables. Given an ordinary, scalar-valued polynomial : $P(x)\; =\; \backslash sum\_^n\; =a\_0\; +\; a\_1\; x+\; a\_2\; x^2\; +\; \backslash cdots\; +\; a\_n\; x^n,$ this polynomial evaluated at a matrix ''A'' is :$P(A)\; =\; \backslash sum\_^n\; =a\_0\; I\; +\; a\_1\; A\; +\; a\_2\; A^2\; +\; \backslash cdots\; +\; a\_n\; A^n,$ where ''I'' is the identity matrix. A matrix polynomial equation is an equality between two matrix polynomials, which holds for the specific matrices in question. A matrix polynomial identity is a matrix polynomial equation which holds for all matrices ''A'' in a specified matrix ring ''Mn''(''R''). == Characteristic and minimal polynomial == The characteristic polynomial of a matrix ''A'' is a scalar-valued polynomial, defined by $p\_A(t)\; =\; \backslash det\; \backslash left(tI\; -\; A\backslash right)$. The Cayley–Hamilton theorem states that if this polynomial is viewed as a matrix polynomial and evaluated at the matrix ''A'' itself, the result is the zero matrix: $p\_A(A)\; =\; 0$. The characteristic polynomial is thus a polynomial which annihilates ''A''. There is a unique monic polynomial of minimal degree which annihilates ''A''; this polynomial is the minimal polynomial. Any polynomial which annihilates ''A'' (such as the characteristic polynomial) is a multiple of the minimal polynomial. It follows that that given two polynomials ''P'' and ''Q'', we have $P(A)\; =\; Q(A)$ if and only if :$P^(\backslash lambda\_i)\; =\; Q^(\backslash lambda\_i)\; \backslash qquad\; \backslash text\; j\; =\; 0,\backslash ldots,n\_i\; \backslash text\; i\; =\; 1,\backslash ldots,s,$ where $P^$ denotes the ''j''th derivative of ''P'' and $\backslash lambda\_1,\; \backslash dots,\; \backslash lambda\_s$ are the eigenvalues of ''A'' with corresponding indices $n\_1,\; \backslash dots,\; n\_s$ (the index of an eigenvalue is the size of its largest Jordan block). 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 polynomial」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.