In matrix theory, Sylvester's formula or Sylvester's matrix theorem (named after J. J. Sylvester) or LagrangeâÂÂSylvester interpolation expresses an analytic function of a matrix as a polynomial in , in terms of the eigenvalues and eigenvectors of . It states that
where the are the eigenvalues of , and the matrices
are the corresponding Frobenius covariants of , which are (projection) matrix Lagrange polynomials of .
Sylvester's formula applies for any diagonalizable matrix with distinct eigenvalues, <sub>1</sub>, ..., <sub>k</sub>, and any function defined on some subset of the complex numbers such that is well defined. The last condition means that every eigenvalue is in the domain of , and that every eigenvalue with multiplicity <sub>i</sub> > 1 is in the interior of the domain, with being () times differentiable at .
Consider the two-by-two matrix:
This matrix has two eigenvalues, 5 and âÂÂ2. Its Frobenius covariants are
Sylvester's formula then amounts to
For instance, if is defined by , then Sylvester's formula expresses the matrix inverse as
Sylvester's formula is only valid for diagonalizable matrices; an extension due to Arthur Buchheim, based on Hermite interpolating polynomials, covers the general case:
where .
A concise form is further given by Hans Schwerdtfeger,
where <sub>i</sub> are the corresponding Frobenius covariants of
If a matrix is both Hermitian and unitary, then it can only have eigenvalues of , and therefore , where is the projector onto the subspace with eigenvalue +1, and is the projector onto the subspace with eigenvalue ; By the completeness of the eigenbasis, . Therefore, for any analytic function ,
In particular, and .