•Proof.(Ü ) By assumption, there is a matrix X12ÎRnxr such that X-Y-1=X12X12*. Defining X2:=Ir completes the construction.
(Þ ) Using Schur complements,
• Y=X-1+X-1X12(X2-X12*X-1 X12)-1 X12* X-1
• Inverting, using the matrix inversion lemma, gives
• Y-1=X-X12*X-1 X12*
• Hence, X-Y-1=X12
X2-1X12*³0, and indeed,
• rank(X-Y-1)=rank(X12 X2-1X12)£ r.