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$tex: J(\mathbf{w})=\frac{\mathbf{w^{\mathrm{T}}S_{\mathrm{B}}w}}{\mathbf{w}^{\mathrm{T}}\mathbf{S}_{\mathrm{W}}\mathbf{w}}$ ..... Eq(1)

Here

$tex: \mathbf{S}_{B}=(\mathbf{m}_{2}-\mathbf{m}_{1})(\mathbf{m}_{2}-\mathbf{m}_{1})^{T}$ ..... Eq(2)

is the between-class covariance matrix.

And

$tex: S_{W}=\sum_{n\epsilon C_{1}}(x_{n}-m_{1})(x_{n}-m_{1})^{T}+\sum_{n\epsilon C_{2}}(x_{n}-m_{2})(x_{n}-m_{2})^{T}$ ..... Eq(3)

is the total within-class covariance matrix.

Differentiating Eq(1) with respect to $tex: \mathbf{w}$ , we find that $tex: J(\mathbf{w})$ is maximized when

$tex: \left(w^{T}S_{B}w\right)S_{W}w=\left(w^{T}S_{W}w\right)S_{B}w$ ..... Eq(4)

From Eq(2), we see that $tex: S_{B}w$ is always in the direction of $tex: (\mathbf{m}_{2}-\mathbf{m}_{1})$ . Furthermore, we do not care about the magnitude of $tex: \mathbf{w}$ , only its direction, and so we can drop the scale factors $tex: \left(w^{T}S_{B}w\right)$ and $tex: \left(w^{T}S_{W}w\right)$ . Multiplying both sides of Eq(4) by $tex: S_{W}^{-1}$ , we obtain

$tex: w\,\alpha\, S_{W}^{-1}(m_{2}-m_{1}).$ ..... Eq(5)

Notice that if the within-class covariance matrix is isotropic, so that $tex: S_{W}$ is proportional to the identity matrix, then $tex: \mathbf{w}$ is proportional to the difference between the class means.