Proof:

The proof is a direct computation:

$$\begin{eqnarray*}&&\left(A+UV^T\right)\left(A^{-1}-A^{-1}U\left(I+V^TA^{-1}U\rig... ...TA^{-1}U\right) \left(I+V^TA^{-1}U\right)^{-1}V^TA^{-1}\\ &=&I. \end{eqnarray*}$$

QED

I now apply the Sherman-Morrison-Woodbury formula to the matrix Jdefined by (9) by defining

$$U=\left[\delta y\vert\gamma Hs\right],\ V=\left[L^Ts\vert L^Ts\right],$$

where

$$\delta=\frac{1}{\alpha s\cdot Hs}>0,\ \gamma=-\frac{1}{s\cdot Hs}<0.$$

Then J=L+UV^T, and, assuming L is nonsingular, J is nonsingular provided

I+V^TL^-1U

is nonsingular. But

$$I+V^TL^{-1}U=\left[\begin{array}{cc} 1+\delta s\cdot y&\gamma... ...}&-1\\ \frac{s\cdot y}{\alpha s\cdot Hs}&0\end{array}\right],$$

and so

$$\det\left(I+V^TL^{-1}U\right)=\frac{s\cdot y}{\alpha s\cdot Hs}>0.$$

Therefore I+V^TL^-1U is nonsingular and hence so is J.