# 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]

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

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.