Note:

The above derivation does not guarantee that J(and hence H₊) is nonsingular. However, I can prove directly that (9) defines a nonsingular matrix J using the following theorem:

Theorem 3.1 (Sherman-Morrison-Woodbury)   Suppose $A\in{\bf {\rm R}}^{n\times n}$ is nonsingular and $U,V\in{\bf {\rm R}}^{n\times p}$ are such that

$$I+V^TA^{-1}U\in{\bf {\rm R}}^{p\times p}$$

is nonsingular. Then

B=A+UV^T

is nonsingular and

$$B^{-1}=A^{-1}-A^{-1}U\left(I+V^TA^{-1}U\right)^{-1}V^TA^{-1}.$$

(Note: The rank of UV^T is easily shown to be p or less.)