Completion to Matrices with Band Inverses and with Minimal Ranks

Abstract

Here we study properties of the unique Green completion obtained in Theorem 8.2 of the previous Chapter 8. In the first section it is shown that this completion is invertible if and only if all the matrices $D_{k}=\tilde{A}(k-n:k,k-n:k),\qquad k=n+1,....,N$ are invertible. In this case all the principal leading submatrices $A^{(j,k)}=A(j:k,j:k),\quad 1\leq j <j+n\leq k\leq N$ of the completion are also invertible.