+The second principal component is w2 n 1 A12z1 A22z2 ### Ak2zk ZX n k a2 k 1 with

+The second principal component is w2

ðn · 1Þ

¼ A12z1 þ A22z2 þ###þ Ak2zk

¼ ZX

ðn · kÞ

a2

ðk · 1Þ

with variance S2 W2 ¼ a0 2RXX a2 We need to maximize this variance subject to the normalizing constraint a0 2a2 ¼ 1 and the orthogonality constraint w0 1w2 ¼ 0. Show that the orthogonality constraint is equivalent to a0 1a2 ¼ 0. Then, using two Lagrange multipliers, one for the normalizing constraint and the other for the orthogonality constraint, show that a2 is an eigenvector corresponding to the second-largest eigenvalue of RXX . Explain how this procedure can be extended to derive the remaining k ' 2 principal components.