Sylvester's formula

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]

f(A)=\sum _{i=1}^{k}f(\lambda _{i})~A_{i}~,

where the $λ i$ are the eigenvalues of $A$ , and the matrices

A_{i}\equiv \prod _{j=1 \atop j\neq i}^{k}{\frac {1}{\lambda _{i}-\lambda _{j}}}\left(A-\lambda _{j}I\right)

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

^ / 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.
^ 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.