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 in terms of the eigenvalues and eigenvectors of A. It states that
Sylvester's formula (1883) is only valid for diagonalizable matrices; an extension due to A. Buchheim (1886) covers the general case.
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.:Def.6.4.
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