13.2.5 The matrix inversion formula

  • Let us consider the four matrices $F$, $G$, $H$ and $K$ and the matrix $F+GHK$. Assume that the inverses of the matrices $F$, $G$ and $(F+GHK)$ exist. Then

    B.5.1\begin{equation} \left( F+GHK \right) ^{-1}= F^{-1}-F^{-1}G \left(H^{-1}+K F^{-1} G\right)^{-1} K F^{-1} \end{equation}

  • Consider the case where $F$ is a $[n \times n]$ square nonsingular matrix, $G=z$ where $z$ is a $[n \times 1]$ vector, $K=z^ T$ and $H=1$. Then the formula simplifies to

    [ (F+z z^ T)^{-1}=F^{-1}-\frac{F^{-1} z z^ T F^{-1}}{1+z^ T F^{-1} z} ]

    where the denominator in the right hand term is a scalar.