Block Matrix
M = \begin{bmatrix} A & B \\ C & D \end{bmatrix}There are 2 Schur complement matrix is considered.
S_D = D - C A^{-1} Band
S_A = A - B D^{-1} CHow we can convert original matrix?
If S_D^{-1} is reqular,
M^{-1} =
\begin{bmatrix}
A & B \\
C & D
\end{bmatrix}^{-1}
=
\begin{bmatrix}
A^{-1} + A^{-1} B S_D^{-1} C A^{-1} & -A^{-1} B S_D^{-1} \\
- S_D^{-1} C A^{-1} & S_D^{-1}
\end{bmatrix}Using the Schur complement, it is possible to divide a large matrix into smaller block matrices and find the inverse matrix.
This is more numerically stable than direct calculation and reduces computational costs.
It allows sequential calculations, and can be calculated efficiently even when only a portion of the matrix is changed.
How to deal with the calculation when Schur is singular
When S_D^{-1} is Singular, We can use S_A^{-1}
M^{-1} =
\begin{bmatrix}
S_A^{-1} & - S_A^{-1} B D^{-1} \\
- D^{-1} C S_A^{-1} & D^{-1} + D^{-1} C S_A^{-1} B D^{-1}
\end{bmatrix}
