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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) |

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Stanford Exploration Project

4/29/2001