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The gradient of will assist us in finding the maximum.
First recall the derivative of a quotient:
| |
(2) |

| (3) |

Using (3)
we set to zero the gradient of the quadratic form
getting
which proves the eigenvector assertion I made above.
Now restate the gradient
while distinguishing missing data from known data
and taking the gradient only with respect to the missing.
To exhibit this most clearly,
we write the operator with
its columns separated into two parts,
one for missing data and one for known,

| |
(4) |

Another expression for the gradient offers more insight
and is often more practical during coding:
Let be a square matrix with ones and zeros on
the main diagonal where ones are found at locations of missing
data and zeros otherwise.
Using ,the gradient is
| |
(5) |

** Next:** CODING
** Up:** Claerbout: Eigenvectors for missing
** Previous:** EIGENVECTOR FORMULATION OF MISSING
Stanford Exploration Project

12/18/1997