= Sherman–Morrison formula
{c}
{title2=$(A+uv^T)^{-1}=A^{-1}-A^{-1}uv^TA^{-1}/(1+v^TA^{-1}u)$}
{wiki}
The <Sherman–Morrison formula> computes the <matrix inverse> after a <rank-one matrix update>. If $A$ is an <invertible matrix> and $\delta=1+v^TA^{-1}u\ne0$, then
$$
(A+uv^T)^{-1}
=A^{-1}-\frac{A^{-1}uv^TA^{-1}}{1+v^TA^{-1}u}.
$$
The inverse correction is itself an <outer product>, divided by the <scalar> $\delta$. The <matrix determinant lemma> shows that the nonzero-denominator condition is exactly the condition that the updated <matrix> remains an <invertible matrix>.
For $A=I$, $u=(1,2)^T$, and $v=(3,-1)^T$, we have $\delta=2$. Thus the computed inverse is
$$
\boxed{
\begin{pmatrix}4&-1\\6&-1\end{pmatrix}^{-1}
=I-\frac12\begin{pmatrix}3&-1\\6&-2\end{pmatrix}
=\begin{pmatrix}-1/2&1/2\\-3&2\end{pmatrix}
}.
$$
Multiplication in either order gives the <identity matrix>. The result reuses the original inverse $I$ and the same correction <vectors>, rather than solving independently for all four inverse entries.
Often only the solution of an updated <linear system> is needed. If $Ax_0=b$ and $Az=u$, then
$$
(A+uv^T)x=b
\quad\Longrightarrow\quad
x=x_0-z\,\frac{v^Tx_0}{1+v^Tz},
$$
provided $1+v^Tz\ne0$. The two solves use the same <LU decomposition> of $A$. This expresses the computational benefit without requiring an explicitly stored <matrix inverse>. Strang's https://ocw.mit.edu/courses/18-065-matrix-methods-in-data-analysis-signal-processing-and-machine-learning-spring-2018/988b4c05680f40a5a4b3be7004a32500_XhSk_Lw2X_U.pdf[Lecture 14, transcript pp. 9–10] explains this reuse of the original factorization.
To see why the inverse correction works, first consider $A=I$ and put $s=v^Tu$. The <rank-one matrix> $R=uv^T$ satisfies $R^2=sR$. Expanding
$$
(I+R)\left(I-\frac{R}{1+s}\right)
=I+R-\frac{R}{1+s}-\frac{sR}{1+s}=I
$$
also works in the reverse order. The general case follows by factoring out $A$ and replacing $u$ by $A^{-1}u$. A matrix identity has reduced to cancellation of scalar coefficients.
Historically, Jack Sherman and Winifred J. Morrison studied updating an inverse when a single matrix entry changes in their https://doi.org/10.1214/aoms/1177729893[1950 paper, Adjustment of an Inverse Matrix Corresponding to a Change in One Element of a Given Matrix]{ref}. Such an entry change is the special <rank-one matrix update> $\alpha e_ie_j^T$, with <standard basis> <vectors> $e_i,e_j$. The reference and the earlier 1949 abstract on row or column changes are recorded in https://epubs.siam.org/doi/10.1137/1023004[Henderson and Searle, On Deriving the Inverse of a Sum of Matrices (1981), references 40–41]. This distinguishes the specific original setting from the more general $uv^T$ formula used here.
Back to article page
