# inverse of matrix with small-rank adjustment

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

 $B=A+XRY^{T},$

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

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

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

Title inverse of matrix with small-rank adjustment InverseOfMatrixWithSmallrankAdjustment 2013-03-22 15:46:06 2013-03-22 15:46:06 kshum (5987) kshum (5987) 8 kshum (5987) Theorem msc 15A09