A Note on the Shorted Operator
C. Allen Butler, Thomas D. Morley · SIAM Journal on Matrix Analysis and Applications · 1988
The Schur complement of a partitioned operator \[ A = \begin{bmatrix} A_{11} & A_{12} \\ A_{21} & A_{22} \end{bmatrix} \] is defined by the formula $S ( A ) = A_{11} - A_{12} A_{22}^{ - 1} A_{21} $. In finite dimensions $S ( A )$ is the unique map $c \mapsto d$ defined by the equations $A_{11} c + A_{12} y = d,A_{21} c + A_{22} y = 0$. In infinite dimensions the shorted operator of a positive operator generalizes the Schur complement; however, the above matrix equations no longer hold. We show in what sense the above equations approximately hold. Applications to infinite networks are shown.