Sylvester's formula


In matrix theory, Sylvester's formula or Sylvester's matrix theorem or Lagrange−Sylvester 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 matrix Lagrange polynomials of.

Conditions

Sylvester's formula applies for any diagonalizable matrix with distinct eigenvalues, 1, …, λk, 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 i > 1 is in the interior of the domain, with being times differentiable at.

Example

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 is defined by, then Sylvester's formula expresses the matrix inverse as

Generalization

Sylvester's formula is only valid for diagonalizable matrices; an extension due to A. Buchheim, based on Hermite interpolating polynomials, covers the general case:
where.
A concise form is further given by Schwerdtfeger,
where i are the corresponding Frobenius covariants of