|
In matrix theory, Sylvester's formula or Sylvester's matrix theorem (named after J. J. Sylvester) or Lagrange−Sylvester interpolation expresses an analytic function of a matrix as a polynomial in , in terms of the eigenvalues and eigenvectors of .〔 Roger A. Horn and Charles R. Johnson (1991), ''Topics in Matrix Analysis''. Cambridge University Press, ISBN 978-0-521-46713-1 〕〔Jon F. Claerbout (1976), ''Sylvester's matrix theorem'', a section of ''Fundamentals of Geophysical Data Processing''. (Online version ) at sepwww.stanford.edu, accessed on 2010-03-14. 〕 It states that :: where the are the eigenvalues of , and the matrices ''i'' are the corresponding Frobenius covariants of , which are (projection) matrix Lagrange polynomials of . Sylvester's formula (1883) is only valid for diagonalizable matrices; an extension due to A. Buchheim (1886) covers the general case. == Conditions == Sylvester's formula applies for any diagonalizable matrix with distinct eigenvalues, 1, …, ''λ''''k'', and any function defined on some subset of the complex numbers such that is well defined. The last condition means that every eigenvalue is in the domain of , and that every eigenvalue with multiplicity ''i'' > 1 is in the interior of the domain, with being () times differentiable at .〔 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「Sylvester's formula」の詳細全文を読む スポンサード リンク
|