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

12 2. Properties of Population Principal ComponentsProof. Let β k be the kth column of B; as the columns of A form a basisfor p-dimensional space, we havep∑β k = c jk α j , k =1, 2,...,q,j=1where c jk ,j=1, 2,...,p, k =1, 2,...,q, are appropriately defined constants.Thus B = AC, where C is the (p × q) matrix with (j, k)th elementc jk ,andB ′ ΣB = C ′ A ′ ΣAC = C ′ ΛC, using (2.1.3)p∑= λ j c j c ′ jj=1where c ′ jNowis the jth row of C. Thereforetr(B ′ ΣB) ====p∑λ j tr(c j c ′ j)j=1p∑λ j tr(c ′ jc j )j=1p∑λ j c ′ jc jj=1p∑j=1 k=1q∑λ j c 2 jk. (2.1.6)C = A ′ B,soC ′ C = B ′ AA ′ B = B ′ B = I q ,because A is orthogonal, and the columns of B are orthonormal. Hencep∑ q∑c 2 jk = q, (2.1.7)j=1 k=1and the columns of C are also orthonormal. The matrix C can be thoughtof as the first q columns of a (p × p) orthogonal matrix, D, say.Buttherows of D are orthonormal and so satisfy d ′ j d j =1,j=1,...,p.Astherows of C consist of the first q elements of the rows of D, it follows thatc ′ j c j ≤ 1, j=1,...,p, that isq∑c 2 jk ≤ 1. (2.1.8)k=1