Jolliffe I. Principal Component Analysis (2ed., Springer, 2002)(518s)

3.2. Geometric Properties of Sample Principal Components 35Figure 3.1. Orthogonal projection of a two-dimensional vector onto a one-dimensionalsubspace.NowThusx ′ ix i =(m i + r i ) ′ (m i + r i )n∑r ′ ir i =i=1= m ′ im i + r ′ ir i +2r ′ im i= m ′ im i + r ′ ir i .n∑x ′ ix i −i=1n∑m ′ im i ,so that, for a given set of observations, minimization of the sum of squaredperpendicular distances is equivalent to maximization of ∑ ni=1 m′ i m i.Distancesare preserved under orthogonal transformations, so the squareddistance m ′ i m i of y i from the origin is the same in y coordinates as inx coordinates. Therefore, the quantity to be maximized is ∑ ni=1 y′ i y i. Butn∑n∑y iy ′ i = x ′ iBB ′ x ii=1i=1i=1