# Sherman-Morrison formula

• nikozm
In summary, the Sherman-Morrison formula provides a simple algebraic expression for the inversion of a rank one perturbation of an invertible matrix. It only involves matrix additions and multiplications, assuming that the inverse of the original matrix is already known. Further details and specifics about the matrix and its inverse are needed for a more precise explanation.

#### nikozm

Hi,

I am trying to derive the Sherman-Morrison formula for the following expression:
a*x*(b*AAH+I)-1*y, where a, b are non-negative scalar values. Also, vectors and matrices are represented by lowercase bold typeface and uppercase bold typeface letters, respectively. Also, (.)H and (.)-1 are the Hermitian transposition and inverse operation, respectively.

Any help would be useful.

Last edited:
You need to give more details.

Sherman-Morrison formula deals with the inversion of a rank one perturbation of an invertible matrix ##A##. Assuming that ##A^{-1}## is already computed, the formula gives you simple algebraic expression for the inverse of the perturbation; simple means that it does not involve any other inversions, only (matrix) additions and multiplications.

What do you want here? You should be more specific.
Is your matrix ##A## a "tall" one? What inverse you assume to be known?

## What is the Sherman-Morrison formula?

The Sherman-Morrison formula is a mathematical equation used to efficiently compute the inverse of a matrix when a small change is made to the original matrix.

## When was the Sherman-Morrison formula first introduced?

The formula was first introduced in 1950 by the mathematicians Wallace Sherman and William Morrison.

## What is the purpose of the Sherman-Morrison formula?

The formula is used to solve linear systems of equations and to efficiently compute the inverse of a matrix when a small change is made to the original matrix.

## What are the advantages of using the Sherman-Morrison formula?

The formula is useful in situations where the inverse of a matrix needs to be computed multiple times with small changes to the matrix. It is also more efficient than traditional methods for computing matrix inverses.

## Are there any limitations to using the Sherman-Morrison formula?

The formula is only applicable to square matrices and may not be as accurate for matrices with large changes or highly ill-conditioned matrices. It also has limitations when used with sparse matrices.