Reference no: EM132852909

Let x denote a p-variate random vector with a finite mean vector µ and a finite full rank covariance matrix Σ. Let yk = γ(x - µ) denote the kth principal component of x. Let b ∈ R p such that b^T b = 1.

Assume that b^T x is uncorrelated with the first k -1 principal components of x.

(a) Let b = d1γ1 + . . . + dpγp. Show that di = 0, when i < k.

(b) Show that var(y) ≥ var(b >x).

Be careful with your notation and note that yk != γk.