The inverse propagator (as it appears in the Lagrangian) has the form . One uses the Sherman–Morrison formula to calculate the inverse (satisfying certain time-ordering boundary conditions) of the inverse propagator—or simply the (Feynman) propagator—which is needed to perform any perturbative calculation involving the spin-1 field. Alternative verification Following is an alternate verification of the Sherman–Morrison formula using the easily verifiable identity . Let then . Substituting gives Generalization (Woodbury matrix identity) Given a square invertible matrix , an matrix , and a matrix , let be an matrix such that . Then, assuming is invertible, we have See also The matrix determinant lemma performs a rank-1 update to a determinant. Woodbury matrix identity Quasi-Newton method Binomial inverse theorem Bunch–Nielsen–Sorensen formula Maxwell stress tensor contains an application of the Sherman–Morrison formula. References External links Linear algebra Matrix theory