Under the set up of Theorem 8 42, explain why Theorem 8 42 immediately implies that q j is a unit eigenvector of K associated with its j th largest eigenvalue j 2 j , which therefore is, up to a factor, the j th principal variance Summarizing, we have proved the Fundamental Theorem of Principal Component Analysis A max vT Kv v 2 v 0, vV1 V V j 1 0

Question: Under the set-up of Theorem 8.42, explain why Theorem 8.42 immediately implies that q j is a unit eigenvector of K associated with its j

Under the set-up of Theorem 8.42, explain why

A; = max vT Kv || v ||2 { v = 0,

Theorem 8.42 immediately implies that q_jis a unit eigenvector of K associated with its j^th largest eigenvalue λ_j = σ²_j, which therefore is, up to a factor, the j^th principal variance. Summarizing, we have proved the Fundamental Theorem of Principal Component Analysis.

A; = max vT Kv || v ||2 { v = 0, vV1 = = V. V j - 1 - 0

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