Find w2 maximizing (w2)TSw2 (w2)Tw2 1 (w2)Tw1 =0 g(w2)=(w2)TSw2-Q(w2)Tw2-1)-B(w2)Tw1-0) 0g(w2)/0w=0 Sw2-aw2-Bw1=0 0 -a0-B1☐ =0 wwd(w =w2)TSw1=1(w2)Tw1=0 Sw1=Aiw1 B=0:Sw2-aw2=0 Sw2=aw2 w2 is the eigenvector of the covariance matrix S Corresponding to the 2nd largest eigenvalue 12Find 𝑤2 maximizing 𝑤2 𝑇𝑆𝑤2 𝑤2 𝑇𝑤2 = 1 𝑤2 𝑇𝑤1 = 0 𝑔 𝑤2 = 𝑤2 𝑇𝑆𝑤2 − 𝛼 𝑤2 𝑇𝑤2 − 1 −𝛽 𝑤2 𝑇𝑤1 − 0 𝜕𝑔 𝑤 Τ 2 𝜕𝑤1 2 = 0 𝜕𝑔 𝑤 Τ 2 𝜕𝑤2 2 = 0 … 𝑆𝑤2 − 𝛼𝑤2 − 𝛽𝑤1 = 0 𝑤1 𝑇𝑆𝑤2 − 𝛼 𝑤1 𝑇𝑤2 − 𝛽 𝑤1 𝑇𝑤1 = 0 0 1 = 𝑤2 𝑇𝑆𝑤1 = 𝑤2 𝑇𝑆 = 𝑤1 𝑇 𝑇𝑤1 𝑆𝑤2 𝑇 = 𝜆1 𝑤2 𝑇𝑤1 = 0 𝛽 = 0: 𝑆𝑤2 − 𝛼𝑤2 = 0 𝑆𝑤2 = 𝛼𝑤2 𝑤2 is the eigenvector of the covariance matrix S Corresponding to the 2nd largest eigenvalue 𝜆2 0 𝑆𝑤1 = 𝜆1𝑤1