91Ó°ÊÓ

The equation \(\left[ {\begin{array}{*{20}{c}}{A - s{I_n}}&B\\C&{{I_m}}\end{array}} \right]\left[ {\begin{array}{*{20}{c}}{\bf{x}}\\{\bf{u}}\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}{\bf{0}}\\{\bf{y}}\end{array}} \right]\) can be expressed as

\(\left( {A - s{I_n}} \right){\bf{x}} + B{\bf{u}} = 0\) and \(C{\bf{x}} + {\bf{u}} = {\bf{y}}\).

If the matrix \(\left( {A - s{I_n}} \right)\) is invertible, then by the equation \(\left( {A - s{I_n}} \right){\bf{x}} + B{\bf{u}}\),

\(\begin{array}{c}\left( {A - s{I_n}} \right){\bf{x}} + B{\bf{u}} = 0\\\left( {A - s{I_n}} \right){\bf{x}} = - B{\bf{u}}\\{\bf{x}} = - {\left( {A - s{I_n}} \right)^{ - 1}}B{\bf{u}}\end{array}\)

Substitute the value of \({\bf{x}}\) in the equation \(C{\bf{x}} + {\bf{u}} = {\bf{y}}\).

\(\begin{array}{c}C\left( { - {{\left( {A - s{I_n}} \right)}^{ - 1}}B{\bf{u}}} \right) + {\bf{u}} = {\bf{y}}\\{\bf{y}} = {I_m}{\bf{u}} - C{\left( {A - s{I_n}} \right)^{ - 1}}B{\bf{u}}\\ = \left( {{I_m} - C{{\left( {A - s{I_n}} \right)}^{ - 1}}B} \right){\bf{u}}\end{array}\)

As \({\bf{y}} = W\left( s \right){\bf{u}}\),

\(W\left( s \right) = {I_m} - C{\left( {A - s{I_n}} \right)^{ - 1}}B\).

So, the value of \(W\left( s \right)\) is \({I_m} - C{\left( {A - s{I_n}} \right)^{ - 1}}B\), and it represents the Schur complement of the matrix \(A - s{I_n}\).