# Sylvester's formula

## From Wikipedia, the free encyclopedia

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 *λ*_{i} are the eigenvalues of A, and the matrices

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