The Fisher linear discriminant is a linear transformation to one dimension used to separate two classes of multivariate observations. Let the
class sample means be 
and 
, and let 
be the sum of the two within-class scatter matrices. Fisher's
criterion chooses a nonzero vector 
to maximize
![J(w)=([w^T(mu_1-mu_2)]^2)/(w^TS_Ww).](/images/equations/FisherLinearDiscriminant/NumberedEquation1.svg) |
When 
is a nonsingular matrix, every maximizing direction
is proportional to
 |
The projected scalar 
can then be compared with a threshold to classify an observation.
The method is also called Fisher's linear discriminant and forms the two-class basis
of linear discriminant analysis.
