Consider the vector yi = (yi1, . . . , yip) 0 , whose sample mean is y = 0, and sample variance-covariance matrix is S. Denote the first principal component of y as ?ci = w? 0yi with corresponding eigenvalue ? >? 0. Denote the second principal component as bci = wb 0yi with corresponding eigenvalue ? >b 0. Use this information to answer the following questions:

d) Consider a second vector xi = Pyi , where P is an orthogonal matrix. Using the formula from the eigenvalue problem, provide expressions for two principal components of x. Hint: At some point, you must utilize the two eigenvectors already provided for S

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Consider the vector yi = (ya, . . . ,yip)', whose sample mean is j? = 0, and sample variancecovariance matrix is S. Denote the rst principal component of y as E!- = w'yi with corresponding eigenvalue A > 0. Denote the second principal component as E} = 'yi with corrmpondmg eigenvalue A > 0. Use this information to answer the following questions: a) Discuss how )1 and :'l compare to each other- (2 Marks) b) Derive a formula for the covariance between E and E. (4 Marks) c) Show that when )i 74 X then the covariance between E and E" is zero- (4 Marks) d) Consider a second vector x,- = Pyi, where P is an orthogonal matrix- Using the formula from the eigenvalue problem, provide expressions for two principal components of x. Hint: At some point, you must utilize the two eigenvectors already provided for S