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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 Square root of a 2 by 2 matrix ： ウィキペディア英語版 Square root of a 2 by 2 matrix A square root of a 2 by 2 matrix ''M'' is another 2 by 2 matrix ''R'' such that ''M'' = ''R''2, where ''R''2 stands for the matrix product of ''R'' with itself. In general there can be no, two, four or even an infinitude of square root matrices. In many cases such a matrix ''R'' can be obtained by an explicit formula. A 2 × 2 matrix with two distinct nonzero eigenvalues has four square roots. A positive-definite matrix has precisely one positive-definite square root. Square roots of a matrix of any dimension come in pairs: If ''R'' is a square root of ''M'', then –''R'' is also a square root of ''M'', since (–''R'')(–''R'') = (–1)(–1)(''RR'') = ''R''2 = ''M''. ==One formula== Let〔Levinger, Bernard W.. 1980. “The Square Root of a 2 × 2 Matrix”. Mathematics Magazine 53 (4). Mathematical Association of America: 222–24. doi:10.2307/2689616.()〕〔P. C. Somayya (1997), ''(Root of a 2x2 Matrix )'', ''The Mathematics Education'', Vol.. XXXI, no. 1. Siwan, Bihar State. INDIA〕 :$$ M = \left( \begin A & B \\ C & D \end\right) where ''A'', ''B'', ''C'', and ''D'' may be real or complex numbers. Furthermore, let ''τ = A + D'' be the trace of ''M'', and ''δ = AD - BC'' be its determinant. Let ''s'' be such that ''s''2 = ''δ'', and ''t'' be such that ''t''2 = ''τ'' + 2''s''. That is, :$$ s = \pm\sqrt , \quad \quad t = \pm \sqrt. Then, if ''t'' ≠ 0, a square root of ''M'' is :$$ R = \frac \left( \begin A + s & B \\ C & D + s \end\right). Indeed, the square of ''R'' is :$$ \begin R^2 &=& \displaystyle \frac \left( \begin (A + s)^2 + B C & (A + s)B + B(D + s) \\ C(A + s) + (D + s)C & (D + s)^2 + B C \end\right)\\() \left( \begin A(A + D + 2s) & (A + D + 2s)B \\ C(A + D + 2 s) & D(A + D + 2 s) \end\right) \;=\; M. \end Note that ''R'' may have complex entries even if ''M'' is a real matrix; this will be the case, in particular, if the determinant ''δ'' is negative. Also, note that ''R'' is positive when ''s>0'' and ''t>0''. 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「Square root of a 2 by 2 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.