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

3.2. Geometric Properties of Sample Principal Components 37mation for which B = A q minimizes the distortion in the configuration asmeasured by ‖YY ′ − XX ′ ‖,where‖·‖ denotes Euclidean norm and Y isa matrix with (i, j)th element ỹ ij − ȳ j .Proof.Y = XB, soYY ′ = XBB ′ X and ‖YY ′ − XX ′ ‖ = ‖XBB ′ X ′ − XX ′ ‖.A matrix result given by Rao (1973, p. 63) states that if F is a symmetricmatrix of rank p with spectral decompositionF = f 1 φ 1 φ ′ 1 + f 2 φ 2 φ ′ 2 + ···+ f p φ p φ ′ p,and G is a matrix of rank q