The second principal component is with variance We need to maximize this variance subject to the normalizing constraint a’ 2 a 2 = 1 and the orthogonality constraint w’ 1 w 2 = 0. Show that the...

The second principal component is

with variance

We need to maximize this variance subject to the normalizing constraint a’₂a₂
= 1 and the orthogonality constraint w’₁w₂
= 0. Show that the orthogonality constraint is equivalent to a’₁a₂
= 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 R_XX
. Explain how this procedure can be extended to derive the remaining k ' 2 principal components.