Find w1 maximizing (w1)TSw1 (w1)Tw1=1 S=Cov(x) Symmetric Positive-semidefinite (non-negative eigenvalues) Using Lagrange multiplier [Bishop,Appendix E] g(w1)=(w1)TSw1-a((w1)Tw1-1) 0g(w1)/0w1=0 Sw1-aw1=0 Sw1=aw1 w1:eigenvector (w1)TSw1=a(w1)Tw1 =0 Choose the maximum one w1 is the eigenvector of the covariance matrix S Corresponding to the largest eigenvalue A1 Find 𝑤1 maximizing 𝑤1 𝑇𝑆𝑤1 𝑤1 𝑇𝑤1 = 1 𝑆 = 𝐶𝑜𝑣 𝑥 Symmetric Positive-semidefinite (non-negative eigenvalues) Using Lagrange multiplier [Bishop, Appendix E] 𝑤1 is the eigenvector of the covariance matrix S Corresponding to the largest eigenvalue 𝜆1 𝑔 𝑤1 = 𝑤1 𝑇𝑆𝑤1 − 𝛼 𝑤1 𝑇𝑤1 − 1 𝜕𝑔 𝑤 Τ 1 𝜕𝑤1 1 = 0 𝜕𝑔 𝑤 Τ 1 𝜕𝑤2 1 = 0 … 𝑆𝑤1 − 𝛼𝑤1 = 0 𝑤1 𝑇𝑆𝑤1 = 𝛼 𝑤1 𝑇𝑤1 = 𝛼 Choose the maximum one 𝑆𝑤1 = 𝛼𝑤1 𝑤1 : eigenvector