Sherman–Morrison formula


In mathematics, in particular linear algebra, the Sherman–Morrison formula, named after Jack Sherman and Winifred J. Morrison, computes the inverse of the sum of an invertible matrix and the outer product,, of vectors and. The Sherman–Morrison formula is a special case of the Woodbury formula. Though named after Sherman and Morrison, it appeared already in earlier publications.

Statement

Suppose is an invertible square matrix and are column vectors. Then is invertible iff . In this case,
Here, is the outer product of two vectors and. The general form shown here is the one published by Bartlett.

Proof

To prove that the backward direction is true, we verify the properties of the inverse. A matrix is the inverse of a matrix if and only if.
We first verify that the right hand side satisfies.
To end the proof of this direction, we need to show that in a similar way as above:
Reciprocally, if, then letting, has a non-regular kernel and is therefore not invertible.

Application

If the inverse of is already known, the formula provides a numerically cheap way to compute the inverse of corrected by the matrix . The computation is relatively cheap because the inverse of does not have to be computed from scratch, but can be computed by correcting .
Using unit columns for or, individual columns or rows of may be manipulated and a correspondingly updated inverse computed relatively cheaply in this way. In the general case, where is a -by- matrix and and are arbitrary vectors of dimension, the whole matrix is updated and the computation takes scalar multiplications. If is a unit column, the computation takes only scalar multiplications. The same goes if is a unit column. If both and are unit columns, the computation takes only scalar multiplications.
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. The inverse propagator has the form. One uses the Sherman-Morrison formula to calculate the inverse of the inverse propagator—or simply the 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