Editing Principal component analysis (section)

=== First component ===
In order to maximize variance, the first weight vector '''w'''<sub>(1)</sub> thus has to satisfy
:<math>\mathbf{w}_{(1)}
 = \arg\max_{\Vert \mathbf{w} \Vert = 1} \,\left\{ \sum_i (t_1)^2_{(i)} \right\}
 = \arg\max_{\Vert \mathbf{w} \Vert = 1} \,\left\{ \sum_i \left(\mathbf{x}_{(i)} \cdot \mathbf{w} \right)^2 \right\}</math>

Equivalently, writing this in matrix form gives
:<math>\mathbf{w}_{(1)}
 = \arg\max_{\left\| \mathbf{w} \right\| = 1} \left\{ \left\| \mathbf{Xw} \right\|^2 \right\}
 = \arg\max_{\left\| \mathbf{w} \right\| = 1} \left\{ \mathbf{w}^\mathsf{T} \mathbf{X}^\mathsf{T} \mathbf{X w} \right\}</math>

Since '''w'''<sub>(1)</sub> has been defined to be a unit vector, it equivalently also satisfies
:<math>\mathbf{w}_{(1)} = \arg\max \left\{ \frac{\mathbf{w}^\mathsf{T} \mathbf{X}^\mathsf{T} \mathbf{X w}}{\mathbf{w}^\mathsf{T} \mathbf{w}} \right\}</math>

The quantity to be maximised can be recognised as a [[Rayleigh quotient]]. A standard result for a [[positive semidefinite matrix]] such as '''X'''<sup>T</sup>'''X''' is that the quotient's maximum possible value is the largest [[eigenvalue]] of the matrix, which occurs when '''''w''''' is the corresponding [[eigenvector]].

With '''w'''<sub>(1)</sub> found, the first principal component of a data vector '''x'''<sub>(''i'')</sub> can then be given as a score ''t''<sub>1(''i'')</sub> = '''x'''<sub>(''i'')</sub> ⋅ '''w'''<sub>(1)</sub> in the transformed co-ordinates, or as the corresponding vector in the original variables, {'''x'''<sub>(''i'')</sub> ⋅ '''w'''<sub>(1)</sub>} '''w'''<sub>(1)</sub>.