inverse of matrix with small-rank adjustment

Suppose that an $n\times n$ matrix $B$ is obtained by adding a small-rank adjustment $XR{Y}^T$ to matrix $A$,

$$B=A+XR{Y}^T,$$

where $X$ and $Y$ are $n\times r$ matrices, and $R$ is an $r\times r$ matrix. Assume that the inverse of $A$ is known and $r$ is much smaller than $n$. The following formula for $B^{-1}$ is often useful,

$$B^{-1}=A^{-1}-A^{-1}X{(R^{-1}+Y^T{A}^{-1}X)}^{-1}{Y}^T{A}^{-1}$$

provided that all inverses in the formula exist.

In particular, when $r=1$ and $A=I$, we have

$${(I+x{y}^T)}^{-1}=I-\frac{x{y}^T}{1+y^{T}x}$$

for any $n\times 1$ column vectors $x$ and $y$ such that $1+y^{T}x\ne 0$.

Title	inverse of matrix with small-rank adjustment
Canonical name	InverseOfMatrixWithSmallrankAdjustment
Date of creation	2013-03-22 15:46:06
Last modified on	2013-03-22 15:46:06
Owner	kshum (5987)
Last modified by	kshum (5987)
Numerical id	8
Author	kshum (5987)
Entry type	Theorem
Classification	msc 15A09