..... Eq(1)
Here
..... Eq(2)
is the between-class covariance matrix.
And
..... Eq(3)
is the total within-class covariance matrix.
Differentiating Eq(1) with respect to , we find that
is maximized when
..... Eq(4)
From Eq(2), we see that is always in the direction of
. Furthermore, we do not care about the magnitude of
, only its direction, and so we can drop the scale factors
and
. Multiplying both sides of Eq(4) by
, we obtain
..... Eq(5)
Notice that if the within-class covariance matrix is isotropic, so that is proportional to the identity matrix, then
is proportional to the difference between the class means.