Editing Schur complement (section)

== Application to solving linear equations ==
The Schur complement arises naturally in solving a system of linear equations such as<ref name="Boyd 2004" /> 

<math>
  \begin{bmatrix} A & B \\ C & D \end{bmatrix}\begin{bmatrix} x \\ y \end{bmatrix}
    = \begin{bmatrix} u \\ v \end{bmatrix} 
</math>.

Assuming that the submatrix <math>A</math> is invertible, we can eliminate <math>x</math> from the equations, as follows.

<math>x = A^{-1} (u - By).</math>

Substituting this expression into the second equation yields
: <math>\left(D - CA^{-1}B\right)y = v-CA^{-1}u.</math>

We refer to this as the ''reduced equation'' obtained by eliminating <math>x</math> from the original equation. The matrix appearing in the reduced equation is called the Schur complement of the first block <math>A</math> in <math>M</math>:
: <math>S \ \overset{\underset{\mathrm{def}}{}}{=}\  D - CA^{-1}B</math>.

Solving the reduced equation, we obtain
: <math>y = S^{-1} \left(v-CA^{-1}u\right).</math>

Substituting this into the first equation yields
: <math>x = \left(A^{-1} + A^{-1} B S^{-1} C A^{-1}\right) u - A^{-1} B S^{-1} v. </math>

We can express the above two equation as:
: <math>\begin{bmatrix} x \\ y \end{bmatrix} =
  \begin{bmatrix}
    A^{-1} + A^{-1} B S^{-1} C A^{-1} & -A^{-1} B S^{-1} \\
                     -S^{-1} C A^{-1} &           S^{-1}
  \end{bmatrix}
  \begin{bmatrix} u \\ v \end{bmatrix}.
</math>

Therefore, a formulation for the inverse of a block matrix is:
: <math>
 \begin{bmatrix} A & B \\ C & D \end{bmatrix}^{-1} = 
 \begin{bmatrix}
   A^{-1} + A^{-1} B S^{-1} C A^{-1} & - A^{-1} B S^{-1} \\
                    -S^{-1} C A^{-1} &            S^{-1}
 \end{bmatrix} = \begin{bmatrix}
   I_p   &  -A^{-1}B\\
         &  I_q
 \end{bmatrix}\begin{bmatrix}
   A^{-1}  &  \\
           &            S^{-1}
 \end{bmatrix}\begin{bmatrix}
   I_p        &  \\
    -CA^{-1}  & I_q
 \end{bmatrix}.
</math>

In particular, we see that the Schur complement is the inverse of the <math>2,2</math> block entry of the inverse of <math>M</math>.

In practice, one needs <math>A</math> to be [[condition number|well-conditioned]] in order for this algorithm to be numerically accurate.

This method is useful in electrical engineering to reduce the dimension of a network's equations.  It is especially useful when element(s) of the output vector are zero.  For example, when <math>u</math> or <math>v</math> is zero, we can eliminate the associated rows of the coefficient matrix without any changes to the rest of the output vector.  If <math>v</math> is null then the above equation for <math>x</math> reduces to  <math>x = \left(A^{-1} + A^{-1} B S^{-1} C A^{-1}\right) u</math>, thus reducing the dimension of the coefficient matrix while leaving <math>u</math> unmodified. This is used to advantage in electrical engineering where it is referred to as node elimination or [[Kron reduction]].