Sylvester's formula

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In matrix theory, Sylvester's formula or Sylvester's matrix theorem (named after J. J. Sylvester) or Lagrange−Sylvester interpolation expresses an analytic function f(A) of a matrix A as a polynomial in A, in terms of the eigenvalues and eigenvectors of A.[1][2] It states that[3]

where the λi are the eigenvalues of A, and the matrices

are the corresponding Frobenius covariants of A, which are (projection) matrix Lagrange polynomials of A.

Conditions[edit]

Sylvester's formula applies for any diagonalizable matrix A with k distinct eigenvalues, λ1, …, λk, and any function f defined on some subset of the complex numbers such that f(A) is well defined. The last condition means that every eigenvalue λi is in the domain of f, and that every eigenvalue λi with multiplicity mi > 1 is in the interior of the domain, with f being (mi — 1) times differentiable at λi.[1]:Def.6.4

Example[edit]

Consider the two-by-two matrix:

This matrix has two eigenvalues, 5 and −2. Its Frobenius covariants are

Sylvester's formula then amounts to

For instance, if f is defined by f(x) = x−1, then Sylvester's formula expresses the matrix inverse f(A) = A−1 as

Generalization[edit]

Sylvester's formula is only valid for diagonalizable matrices; an extension due to A. Buchheim, based on Hermite interpolating polynomials, covers the general case:[4]

,

where .

A concise form is further given by Schwerdtfeger,[5]

,

where Ai are the corresponding Frobenius covariants of A

See also[edit]

References[edit]

  1. ^ a b Roger A. Horn and Charles R. Johnson (1991), Topics in Matrix Analysis. Cambridge University Press, ISBN 978-0-521-46713-1
  2. ^ 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.
  3. ^ Sylvester, J.J. (1883). "XXXIX. On the equation to the secular inequalities in the planetary theory". The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science. 16 (100): 267–269. doi:10.1080/14786448308627430. ISSN 1941-5982.
  4. ^ Buchheim, A. (1884). "On the Theory of Matrices". Proceedings of the London Mathematical Society. s1-16 (1): 63–82. doi:10.1112/plms/s1-16.1.63. ISSN 0024-6115.
  5. ^ Schwerdtfeger, Hans (1938). Les fonctions de matrices: Les fonctions univalentes. I, Volume 1. Paris, France: Hermann.