Question

Given a data matrix X as in Question 2, Assume the means of the p variables are zero. Let S = 1X,X be the sample covariance matrix. Let λι > λ2 > . . . > λ, be the ordered eigenvalues of S. Let el, . . . , ep be their corresponding orthogonal eigenvectors with unit length. In multivariate analysis, we usually want to use the first few eigenvalues and eigenvectors to represent the original data, as a tool of dimension reduction On one aspect, let be an approximate of S for m<p. Calculate tm) and trE(S- Sm)2)/tr(S2), where tr(S2) can be regarded as the total variation of the data. o On another aspect, {Xei,..., Xem) are the transformed data by the eigenvectors. Calculate the sample covariance St of (Xei,..., Xem (regard the sample mean as 0). What is tr(S2) comparing to tr(S2)? What can you conclude on the dimension reduction by eigenvectors from the above two points?

Answer 1