PCA Project all the data points x onto w1, and obtain a set of z1 z=Wx We want the variance of z1 as large as possible Reduce to 1-D: Vare)=∑a-a2w2=1 Z1=w1·x Z1 Z2 w2.x We want the variance of z2 as large as [(w1)T possible W = (w2)T var(z)=∑a,-a2w2l2=1 w1w2=0 Orthogonal matrixPCA We want the variance of 𝑧2 as large as possible 𝑉𝑎𝑟 𝑧2 = 1 𝑁 ෍ 𝑧2 𝑧2 − 𝑧ഥ2 2 𝑤2 2 = 1 𝑤1 ∙ 𝑤2 = 0 𝑧 = 𝑊𝑥 𝑧1 = 𝑤1 ∙ 𝑥 Reduce to 1-D: Project all the data points x onto 𝑤1 , and obtain a set of 𝑧1 We want the variance of 𝑧1 as large as possible 𝑉𝑎𝑟 𝑧1 = 1 𝑁 ෍ 𝑧1 𝑧1 − 𝑧ഥ1 2 𝑤1 2 = 1 𝑧2 = 𝑤2 ∙ 𝑥 𝑊 = 𝑤1 𝑇 𝑤2 𝑇 ⋮ Orthogonal matrix