Here is the formula about PCA (maximize the projected variance). My confusion is how to derive from formula 12.2 to formula 12.3?
2026-03-30 07:12:30.1774854750
PCA -- principal component analysis formula issue
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\begin{align} \left( \mathbf u_1^T \mathbf x_n - \mathbf u_1^T \bar{\mathbf x} \right)^2 &= \left( \mathbf u_1^T ( \mathbf x_n - \bar{\mathbf x} ) \right)^2 \\ &= \mathbf u_1^T ( \mathbf x_n - \bar{\mathbf x} ) \cdot \mathbf u_1^T ( \mathbf x_n - \bar{\mathbf x} ) \\ &= \mathbf u_1^T ( \mathbf x_n - \bar{\mathbf x} ) \cdot ( \mathbf x_n - \bar{\mathbf x} )^T \mathbf u_1 \\ &= \mathbf u_1^T \cdot \underbrace{( \mathbf x_n - \bar{\mathbf x} ) ( \mathbf x_n - \bar{\mathbf x} )^T}_{n\text{-th term of }\mathbf S} \cdot \mathbf u_1 \end{align}