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Let us define the block matrix **M** as follows:
| |
(87) |

where **A**, **B**, **C**, and **D** are matrices.
First, we consider the matrix equation
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(88) |

If we multiply the top row by and add it to the bottom,
we have
| |
(89) |

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
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(90) |

which resembles an *LDU* decomposition of **M**.
Alternatively, we have the *UDL* decomposition
| |
(91) |

where is the Schur complement of
**D**.
The inversion formulas are then easy to derive as follows:
| |
(92) |

and
| |
(93) |

The decomposition of the matrix **M** offers opportunities for
fast inversion algorithms. The final expressions for **M** are
| |
(94) |

and
| |
(95) |

Equations () and () yield the matrix inversion lemma
| |
(96) |

** Next:** Inversion of the Hessian
** Up:** Least-squares solution of the
** Previous:** Least-squares solution of the
Stanford Exploration Project

5/5/2005