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Let us define the block matrix M as follows:
| |
(18) |
where A, B, C, and D are matrices.
First, we consider the matrix equation
| |
(19) |
If we multiply the top row by and add it to the bottom,
we have
| |
(20) |
Then we can easily find F and E. The quantity is called the Schur complement of A and,
denoted as , appears often in linear algebra Demmel (1997).
The derivation of F and E can be written in a matrix form
| |
(21) |
which resembles an LDU decomposition of M.
Alternatively, we have the UDL decomposition
| |
(22) |
where is the Schur complement of
D.
The inversion formulas are then easy to derive as follows:
| |
(23) |
and
| |
(24) |
The decomposition of the matrix M offers opportunities for
fast inversion algorithms. The final expressions for M are
| |
(25) |
and
| |
(26) |
Equations (25) and (26) yield the matrix inversion lemma
| |
(27) |
Next: Inversion of the Hessian
Up: Appendix
Previous: Appendix
Stanford Exploration Project
4/29/2001