Math 6610 • Analysis of Numerical Methods I

{Sherman-Morrison-Woodbury Formula} Suppose \(A\) is an invertible
\(n\times n\) matrix, and \(U\) and \(V\) are \(n\times k\) matrices such that
\(A+UV^T\) and the \(k\times k\) matrix \(I+V^TA^{-1}U\) are invertible. Show that
\[
  (A+UV^T)^{-1}
  =
  A^{-1}-A^{-1}U(I+V^TA^{-1}U)^{-1}V^TA^{-1}
  \tag{2}
\]
Explain why this means that a rank \(k\) change of \(A\) leads to a rank \(k\) change
of \(A^{-1}\).