Singular value decomposition let x be a n x p data matrix

Question

Using Eq. (3.103a and b) prove the results of Theorem 3.17.

THEOREM 3.17

(Singular Value Decomposition) Let X be a (n x p) data matrix such that p(X) = p ≤n, = 0, and let X^TX and XX^T be (p x p) and (n x n) Grammian matrices respectively, Then

(i) Nonzero latent vectors of XX^T are standardized PCs of X^TX.

(ii) Nonzero latent roots of X^TX and XX^T rare equal.

(iii) Any real (n x p) matrix X can be decomposed as