**Bob Fisher**

The Fisher Linear Discriminant (FLD) gives a projection matrix W that reshapes the scatter of a data set to maximize class separability, defined as the ratio of the between-class scatter matrix to the within-class scatter matrix. This projection defines features that are optimally discriminating.

Let be a set of *N* column vectors of dimension *D*.
The mean of the dataset is

There are *K* classes .
The mean of class *k* containing members is:

The between class scatter matrix is

The within class scatter matrix is

The transformation matrix that repositions the data
to be most separable is the matrix W
that maximizes

Let
be the generalized eigenvectors of
and .
Then .
This gives a projection space of dimension *D*.
A projection space of dimension *d* < *D* can be defined
by using the generalized eigenvectors with the largest *d* eigenvalues
to give .
The projection of vector into a subspace of dimension *d*
is .

The generalized eigenvectors are eigenvectors of

Friday June 15 18:23:13 BST 2001