Sherman-Morrison formula

Definition

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

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

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