I thenbspith minimum angle between an arbitrary vector in

Question

Prove the second part of Theorem 5.10.

THEOREM 5.10

(Krzanowski, 1979) Let

and U=P₍₁₎H where U is a unit vector, M is a diagonal matrix of r nonzero latent roots, and H represents orthogonal latent vectors.

(i) The ith minimum angle between an arbitrary vector in the space of the first r principal components of the first sample, and the one most nearly parallel to it in the space of the first r components of the second sample is cos^-1(m_i^1/2) where m_i is the ith latent root of (P^T₍₁₎P₍₂₎)(P₍₁₎P₍₂₎)^T