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 invertible with inverse given as above) is true, we verify the properties of the inverse. A matrix (in this case the right-hand side of the Sherman–Morrison formula) is the inverse of a matrix (in this case ) 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: (In fact, the last step can be avoided since for square matrices and , is equivalent to .) () Reciprocally, if , then via the matrix determinant lemma, , so is 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 (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) . Using unit columns (columns from the identity matrix) 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.