Let X = [X X2 eigenvector/eigenvalue Xp] be a random vector with covariance matrix . The...

  

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Let X = [X₁ X2 eigenvector/eigenvalue Xp] be a random vector with covariance matrix Σ. The pairs of Σ are given by (e₁, A;) for i = 1,..., p. The ith principal component is given by Y; = eX. The eigenmatrix of is given by ... E = [₁ e₂ ep], and corresponding diagonal eigenvalue matrix is given by: so that: A 0 0 1₂ : : 0 0 ... ... ... ... Σ = ΕΛΕ΄. 0 0 : (b) Show that the principal components exactly decompose the total population variance, i.e. tr(2) = tr(A). Hint: you may find it useful to know that tr(AB) = tr(BA). Activa [3 marks] to S Let X = [X₁ X2 eigenvector/eigenvalue Xp] be a random vector with covariance matrix Σ. The pairs of Σ are given by (e₁, A;) for i = 1,..., p. The ith principal component is given by Y; = eX. The eigenmatrix of is given by ... E = [₁ e₂ ep], and corresponding diagonal eigenvalue matrix is given by: so that: A 0 0 1₂ : : 0 0 ... ... ... ... Σ = ΕΛΕ΄. 0 0 : (b) Show that the principal components exactly decompose the total population variance, i.e. tr(2) = tr(A). Hint: you may find it useful to know that tr(AB) = tr(BA). Activa [3 marks] to S

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Answer Lets denote the total population variance by tr and the trace of the diagonal matrix o... View the full answer