Sherman-Morrison formula

Definition

Suppose $n$ is a positive integer, $A$ is an invertible $n\times n$ matrix, and $\overrightarrow{u},\overrightarrow{v}$ are $n$-dimensional column vectors. The Hadamard product $\overrightarrow{u}{\overrightarrow{v}}^T$ is therefore a $n\times n$ matrix of rank one. We have the following:

• The matrix $A+\overrightarrow{u}{\overrightarrow{v}}^T$ is invertible if and only if the real number $1+{\overrightarrow{v}}^T{A}^{-1}\overrightarrow{u}$ is nonzero.
• If the condition above holds, we have the formula:

$(A+\overrightarrow{u}{\overrightarrow{v}}^T{)}^{-1}=A^{-1}-\frac{A^{-1}\overrightarrow{u}{\overrightarrow{v}}^T{A}^{-1}}{1+{\overrightarrow{v}}^T{A}^{-1}\overrightarrow{u}}$