If \(A\) is any \(n \times n\) matrix, let \(^{(r)}A\) denote the \(r \times r\) submatrix in the upper left corner of \(A\); that is, \(^{(r)}A\) is the matrix obtained from \(A\) by deleting the las...If \(A\) is any \(n \times n\) matrix, let \(^{(r)}A\) denote the \(r \times r\) submatrix in the upper left corner of \(A\); that is, \(^{(r)}A\) is the matrix obtained from \(A\) by deleting the last \(n - r\) rows and columns. \[U^TU = (U_{1}^TD_{2}^{-1})(D_{2}^{-1}U_{1}) = U_{1}^T(D_{2}^2)^{-1}U_{1} = (U_{1}^TD_{1}^{-1})U_{1} = (L_{1}^{-1})U_{1} = A \nonumber \]
