15.1 PCA and maximal variance. Let X be an uncentered data matrix and let

x = 1 m

P i xi be the sample mean of the columns of X.

(a) Show that the variance of one-dimensional projections of the data onto an arbitrary vector u equals u>Cu, where C = 1 m

P i(xi ???? x)(xi ???? x)> is the sample covariance matrix.

(b) Show that PCA with k = 1 projects the data onto the direction (i.e., u>u =

1) of maximal variance.