Enter the covariance matrix S into R and call the matrix S. (1) b. Calculate the total sample...

Question

Enter the covariance matrix S into R and call the matrix S. (1) b. Calculate the total sample variance of S (without using eigenvalues). (1) c. Calculate the generalized sample variance of S (without using eigenvalues). (1) d. Find the eigenvalues and eigenvectors of S. (2) e. Use the eigenvalues to compute the total sample variance of S. (1) f. Use the eigenvalues to compute the generalized sample variance of S. (1) g. Define a matrix P with columns containing the eigenvectors of S. Display P. (1) h. Show that P is an orthogonal matrix. (2) i. Find the sample covariance matrix of Y (say Sy) where Y P X = ' and 1 2 3 X  =[ , , ] . X X X Hint: You can get this from the covariance matrix S. Display Sy, rounded to three decimal places. (3) j. Looking at Sy, what can be said about the components of Y ? (1) k. Compute the trace of S and the trace of Sy. What do you notice?

Monica · Accepted Answer

> #a
> S  print(S)
     [,1] [,2] [,3]
[1,]    3   -1    2
[2,]   -1    5    1
[3,]    2    1    7
> #b
> total_sample_variance  print(total_sample_variance)
[1] 5
> #c
> generalized_sample_variance  print(generalized_sample_variance)
[1] 2.111111
> #d
> eigen_output  print(eigen_output)
eigen() decomposition
$values
[1] 7.939235 5.406728 1.654037
$vectors
          [,1]        [,2]       [,

Enter the covariance matrix S into R and call the matrix S. (1) b. Calculate the total sample variance of S (without using eigenvalues). (1) c. Calculate the generalized sample variance of S (without...

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