In linear algebra, the Sherman–Morrison formula, named after Jack Sherman and Winifred J. Morrison, computes the inverse of a "rank-1 update" to a matrix whose inverse has previously been computed.
the formula cheaply computes an updated matrix inverse
Though named after Sherman and Morrison, it appeared already in earlier publications.
is an invertible square matrix and
is the outer product of two vectors
The general form shown here is the one published by Bartlett.
) To prove that the backward direction
(in this case the right-hand side of the Sherman–Morrison formula) is the inverse of a matrix
We first verify that the right hand side (
To end the proof of this direction, we need to show that
in a similar way as above: (In fact, the last step can be avoided since for square matrices
, then via the matrix determinant lemma,
is already known, the formula provides a numerically cheap way to compute the inverse of
(depending on the point of view, the correction may be seen as a perturbation or as a rank-1 update).
The computation is relatively cheap because the inverse of
does not have to be computed from scratch (which in general is expensive), but can be computed by correcting (or perturbing)
, individual columns or rows of
may be manipulated and a correspondingly updated inverse computed relatively cheaply in this way.
are arbitrary vectors of dimension
, the whole matrix is updated[5] and the computation takes
is a unit column, the computation takes only
are unit columns, the computation takes only
This formula also has application in theoretical physics.
Namely, in quantum field theory, one uses this formula to calculate the propagator of a spin-1 field.
[8][circular reference] 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[9] involving the spin-1 field.
One of the issues with the formula is that little is known about its numerical stability.
There are no published results concerning its error bounds.
Anecdotal evidence[10] suggests that the Woodbury matrix identity (a generalization of the Sherman–Morrison formula) may diverge even for seemingly benign examples (when both the original and modified matrices are well-conditioned).
Following is an alternate verification of the Sherman–Morrison formula using the easily verifiable identity Let then Substituting