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. It states that

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

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

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

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