Consider a data set with 20 two-dimensional points of the form (i. i). (i.i 1) for i, 10 1. Center the data set by showing the new coordinates of (i, i) and (i, i +1) 2, Compute the sample covariance matrix of the centered data S-n1XX, where X is the 20 x 2 matrix for the centered data. 3. Find the two principal components ul, . Each should be a two-dimension vector with norm 1. Hint you may perform eigendecomposition online with http://www.bluebit.gr/matrix-calculator/. Here is an example to interpret the result. If the input matrix is (this is just an example, not your S) 28.000 13.856 13.856 12.000 the program will return Eigenvalues (36.000, 0.000i) 4.000, 0.000i) Eigenvectors: ( 0 . 866 , 0 . 0001) (-0. 500 , 0 . 000i) 0.500, 0.0001) 0.866, 0.000i) Ignore the imaginary part. The format means the leading eigenvalue is 36, the corresponding eigen vector is the first column, namely (0.866,0.500)T; the second eigenvalue is 4, and the corresponding eigenvector is (-0.500,0.866) 4. Compute the new two-dimensional coordinates of CENTERED points - corresponding to the old (i, i) and (i, i+1)-by projecting them onto ul,U2. (Hint: if you only keep the first dimension, it would be a maximum spread projection of the original data set.)