In matrix theory, Sylvester's formula or Sylvester's matrix theorem (named after J. J. Sylvester) or Lagrange−Sylvester interpolation expresses an analytic function of a matrix as a polynomial in , in terms of the eigenvalues and eigenvectors of ./ Roger A. Horn and Charles R. Johnson (1991), Topics in Matrix Analysis. Cambridge University Press, Jon F. Claerbout (1976), Sylvester's matrix theorem, a section of Fundamentals of Geophysical Data Processing. Online version at sepwww.stanford.edu, accessed on 2010-03-14. 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 .

Conditions

Sylvester's formula applies for any diagonalizable matrix with distinct eigenvalues, 1, ..., k, 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 matrices with no repeated eigenvalues; an extension due to Arthur Buchheim, based on Hermite interpolating polynomials, covers the general case: , where .

a concise form of the generalized Sylvester's formula is further given by Hans Schwerdtfeger, , where i are the corresponding generalized Frobenius covariants of

Special case

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 .

See also Adjugate matrix Holomorphic functional calculus Resolvent formalism

References

F.R. Gantmacher, The Theory of Matrices v I (Chelsea Publishing, NY, 1960) , pp 101-103