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]

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

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